Convergence to a fixed point












3












$begingroup$


Let $f : [a,b] rightarrow [a,b]$ be a continuous function s.t. $f'(x)$ is defined on $(a,b)$ and $leftlvert f'(x)rightrvert leqq t$ where $0<t<1$. Prove that for any point $x_0$ in $[a,b]$ the sequence defined by $$ x_n=f(x_{n-1}), n>0$$
converges to one unique fixed point.



Attempt:
Frankly, I have struggled to make a real attempt due to the fact that I can't find notes relating to this.



Obviously, I am assuming that there exists $x$ in$ [a,b]$ s.t. $f(x)=x$ but how do I relate the sequence to this $x$?



I'm strictly not allowed to assume Banach's theorem in this question, nor the Cauchy sequence because they come up on the second part of the course. I rather have to PROVE this.










share|cite|improve this question









New contributor




Grace is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$












  • $begingroup$
    Another related question.
    $endgroup$
    – rtybase
    4 hours ago






  • 1




    $begingroup$
    Possible duplicate of Contraction mapping in the context of $f(x_n)=x_{n+1}$.
    $endgroup$
    – rtybase
    4 hours ago










  • $begingroup$
    Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
    $endgroup$
    – dantopa
    2 hours ago


















3












$begingroup$


Let $f : [a,b] rightarrow [a,b]$ be a continuous function s.t. $f'(x)$ is defined on $(a,b)$ and $leftlvert f'(x)rightrvert leqq t$ where $0<t<1$. Prove that for any point $x_0$ in $[a,b]$ the sequence defined by $$ x_n=f(x_{n-1}), n>0$$
converges to one unique fixed point.



Attempt:
Frankly, I have struggled to make a real attempt due to the fact that I can't find notes relating to this.



Obviously, I am assuming that there exists $x$ in$ [a,b]$ s.t. $f(x)=x$ but how do I relate the sequence to this $x$?



I'm strictly not allowed to assume Banach's theorem in this question, nor the Cauchy sequence because they come up on the second part of the course. I rather have to PROVE this.










share|cite|improve this question









New contributor




Grace is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$












  • $begingroup$
    Another related question.
    $endgroup$
    – rtybase
    4 hours ago






  • 1




    $begingroup$
    Possible duplicate of Contraction mapping in the context of $f(x_n)=x_{n+1}$.
    $endgroup$
    – rtybase
    4 hours ago










  • $begingroup$
    Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
    $endgroup$
    – dantopa
    2 hours ago
















3












3








3





$begingroup$


Let $f : [a,b] rightarrow [a,b]$ be a continuous function s.t. $f'(x)$ is defined on $(a,b)$ and $leftlvert f'(x)rightrvert leqq t$ where $0<t<1$. Prove that for any point $x_0$ in $[a,b]$ the sequence defined by $$ x_n=f(x_{n-1}), n>0$$
converges to one unique fixed point.



Attempt:
Frankly, I have struggled to make a real attempt due to the fact that I can't find notes relating to this.



Obviously, I am assuming that there exists $x$ in$ [a,b]$ s.t. $f(x)=x$ but how do I relate the sequence to this $x$?



I'm strictly not allowed to assume Banach's theorem in this question, nor the Cauchy sequence because they come up on the second part of the course. I rather have to PROVE this.










share|cite|improve this question









New contributor




Grace is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




Let $f : [a,b] rightarrow [a,b]$ be a continuous function s.t. $f'(x)$ is defined on $(a,b)$ and $leftlvert f'(x)rightrvert leqq t$ where $0<t<1$. Prove that for any point $x_0$ in $[a,b]$ the sequence defined by $$ x_n=f(x_{n-1}), n>0$$
converges to one unique fixed point.



Attempt:
Frankly, I have struggled to make a real attempt due to the fact that I can't find notes relating to this.



Obviously, I am assuming that there exists $x$ in$ [a,b]$ s.t. $f(x)=x$ but how do I relate the sequence to this $x$?



I'm strictly not allowed to assume Banach's theorem in this question, nor the Cauchy sequence because they come up on the second part of the course. I rather have to PROVE this.







analysis convergence numerical-methods fixed-point-theorems fixedpoints






share|cite|improve this question









New contributor




Grace is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




Grace is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited 3 hours ago







Grace













New contributor




Grace is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 4 hours ago









Grace Grace

164




164




New contributor




Grace is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Grace is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Grace is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • $begingroup$
    Another related question.
    $endgroup$
    – rtybase
    4 hours ago






  • 1




    $begingroup$
    Possible duplicate of Contraction mapping in the context of $f(x_n)=x_{n+1}$.
    $endgroup$
    – rtybase
    4 hours ago










  • $begingroup$
    Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
    $endgroup$
    – dantopa
    2 hours ago




















  • $begingroup$
    Another related question.
    $endgroup$
    – rtybase
    4 hours ago






  • 1




    $begingroup$
    Possible duplicate of Contraction mapping in the context of $f(x_n)=x_{n+1}$.
    $endgroup$
    – rtybase
    4 hours ago










  • $begingroup$
    Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
    $endgroup$
    – dantopa
    2 hours ago


















$begingroup$
Another related question.
$endgroup$
– rtybase
4 hours ago




$begingroup$
Another related question.
$endgroup$
– rtybase
4 hours ago




1




1




$begingroup$
Possible duplicate of Contraction mapping in the context of $f(x_n)=x_{n+1}$.
$endgroup$
– rtybase
4 hours ago




$begingroup$
Possible duplicate of Contraction mapping in the context of $f(x_n)=x_{n+1}$.
$endgroup$
– rtybase
4 hours ago












$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
$endgroup$
– dantopa
2 hours ago






$begingroup$
Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
$endgroup$
– dantopa
2 hours ago












3 Answers
3






active

oldest

votes


















2












$begingroup$

The Mean Value Theorem tells you that the sequence ${x_n }$ is Cauchy because $|f(y)-f(x)| leq t|y-x|$. The space $[0, 1]$ is complete, so since the sequence is Cauchy, it must converge.






share|cite|improve this answer











$endgroup$





















    1












    $begingroup$

    This can be proved using the Banach Fixed Point Theorem.



    Intuitively, the BFPT tells us that if there is some function $F$ such that the distance between any two points $x$ and $y$ (when scaled by a constant $q$) is always larger than the distance between the corresponding images ($F(x)$ and $F(y)$), then the sequence
    $$x_n = F(x_{n-1}) $$
    will converge to a unique fixed point. For a more rigorous treatment of the BFPT statement: https://en.wikipedia.org/wiki/Banach_fixed-point_theorem.



    Since in this case you know that
    $$|f'(x)| leq t $$
    This implies that



    $$Big|frac{f(x) - f(y)}{x - y}Big| leq t$$



    for all possible points $x,y in [a,b]$. Think about why this is the case (Hint: Use the mean value theorem).



    This implies that



    $$| f(x) - f(y)| < t |x-y| $$



    which is what the BFPT requires. From here, we can just apply the BFPT to state that there is a fixed point in $[a,b]$ for the sequence



    $$ x_n = f(x_{n-1})$$






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Should probably expressly mention that the space $[0, 1]$ is complete, since that is a necessary condition for the Banach fixed point theorem to hold. For an easy counterexample demonstrating that you need completeness, consider $f(x) = x/2$ on $(0, 1]$.
      $endgroup$
      – Robert Shore
      4 hours ago










    • $begingroup$
      @RobertShore Yes, you are right, thanks!
      $endgroup$
      – Sean Lee
      4 hours ago



















    1












    $begingroup$

    To prove this from scratch note that by MVT $$|x_n-x_m| leq |x_{m+1}-x_m|+|x_{m+2}-x_{m+1}|$$ $$+cdots+|x_{n}-x_{n-1}|leq |x_{m+1}-x_m| (1+t+t^{2}+cdots+t^{n+m-1})$$ for $n >m$. Also $|x_{m+1}-x_m| leq t^{m-1} |x_2-x_1|$. Using the convergence of the geometric series $sum t^{n}$ conclude that ${x_n}$ is Cauchy. If $x =lim x_n$ then the definition of $x_n$'s and continuity of $f$ tells you that $f(x)=x$. Uniqueness follows easily by MVT.






    share|cite|improve this answer









    $endgroup$













      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "69"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      autoActivateHeartbeat: false,
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });






      Grace is a new contributor. Be nice, and check out our Code of Conduct.










      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3138183%2fconvergence-to-a-fixed-point%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2












      $begingroup$

      The Mean Value Theorem tells you that the sequence ${x_n }$ is Cauchy because $|f(y)-f(x)| leq t|y-x|$. The space $[0, 1]$ is complete, so since the sequence is Cauchy, it must converge.






      share|cite|improve this answer











      $endgroup$


















        2












        $begingroup$

        The Mean Value Theorem tells you that the sequence ${x_n }$ is Cauchy because $|f(y)-f(x)| leq t|y-x|$. The space $[0, 1]$ is complete, so since the sequence is Cauchy, it must converge.






        share|cite|improve this answer











        $endgroup$
















          2












          2








          2





          $begingroup$

          The Mean Value Theorem tells you that the sequence ${x_n }$ is Cauchy because $|f(y)-f(x)| leq t|y-x|$. The space $[0, 1]$ is complete, so since the sequence is Cauchy, it must converge.






          share|cite|improve this answer











          $endgroup$



          The Mean Value Theorem tells you that the sequence ${x_n }$ is Cauchy because $|f(y)-f(x)| leq t|y-x|$. The space $[0, 1]$ is complete, so since the sequence is Cauchy, it must converge.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 4 hours ago

























          answered 4 hours ago









          Robert ShoreRobert Shore

          2,044116




          2,044116























              1












              $begingroup$

              This can be proved using the Banach Fixed Point Theorem.



              Intuitively, the BFPT tells us that if there is some function $F$ such that the distance between any two points $x$ and $y$ (when scaled by a constant $q$) is always larger than the distance between the corresponding images ($F(x)$ and $F(y)$), then the sequence
              $$x_n = F(x_{n-1}) $$
              will converge to a unique fixed point. For a more rigorous treatment of the BFPT statement: https://en.wikipedia.org/wiki/Banach_fixed-point_theorem.



              Since in this case you know that
              $$|f'(x)| leq t $$
              This implies that



              $$Big|frac{f(x) - f(y)}{x - y}Big| leq t$$



              for all possible points $x,y in [a,b]$. Think about why this is the case (Hint: Use the mean value theorem).



              This implies that



              $$| f(x) - f(y)| < t |x-y| $$



              which is what the BFPT requires. From here, we can just apply the BFPT to state that there is a fixed point in $[a,b]$ for the sequence



              $$ x_n = f(x_{n-1})$$






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                Should probably expressly mention that the space $[0, 1]$ is complete, since that is a necessary condition for the Banach fixed point theorem to hold. For an easy counterexample demonstrating that you need completeness, consider $f(x) = x/2$ on $(0, 1]$.
                $endgroup$
                – Robert Shore
                4 hours ago










              • $begingroup$
                @RobertShore Yes, you are right, thanks!
                $endgroup$
                – Sean Lee
                4 hours ago
















              1












              $begingroup$

              This can be proved using the Banach Fixed Point Theorem.



              Intuitively, the BFPT tells us that if there is some function $F$ such that the distance between any two points $x$ and $y$ (when scaled by a constant $q$) is always larger than the distance between the corresponding images ($F(x)$ and $F(y)$), then the sequence
              $$x_n = F(x_{n-1}) $$
              will converge to a unique fixed point. For a more rigorous treatment of the BFPT statement: https://en.wikipedia.org/wiki/Banach_fixed-point_theorem.



              Since in this case you know that
              $$|f'(x)| leq t $$
              This implies that



              $$Big|frac{f(x) - f(y)}{x - y}Big| leq t$$



              for all possible points $x,y in [a,b]$. Think about why this is the case (Hint: Use the mean value theorem).



              This implies that



              $$| f(x) - f(y)| < t |x-y| $$



              which is what the BFPT requires. From here, we can just apply the BFPT to state that there is a fixed point in $[a,b]$ for the sequence



              $$ x_n = f(x_{n-1})$$






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                Should probably expressly mention that the space $[0, 1]$ is complete, since that is a necessary condition for the Banach fixed point theorem to hold. For an easy counterexample demonstrating that you need completeness, consider $f(x) = x/2$ on $(0, 1]$.
                $endgroup$
                – Robert Shore
                4 hours ago










              • $begingroup$
                @RobertShore Yes, you are right, thanks!
                $endgroup$
                – Sean Lee
                4 hours ago














              1












              1








              1





              $begingroup$

              This can be proved using the Banach Fixed Point Theorem.



              Intuitively, the BFPT tells us that if there is some function $F$ such that the distance between any two points $x$ and $y$ (when scaled by a constant $q$) is always larger than the distance between the corresponding images ($F(x)$ and $F(y)$), then the sequence
              $$x_n = F(x_{n-1}) $$
              will converge to a unique fixed point. For a more rigorous treatment of the BFPT statement: https://en.wikipedia.org/wiki/Banach_fixed-point_theorem.



              Since in this case you know that
              $$|f'(x)| leq t $$
              This implies that



              $$Big|frac{f(x) - f(y)}{x - y}Big| leq t$$



              for all possible points $x,y in [a,b]$. Think about why this is the case (Hint: Use the mean value theorem).



              This implies that



              $$| f(x) - f(y)| < t |x-y| $$



              which is what the BFPT requires. From here, we can just apply the BFPT to state that there is a fixed point in $[a,b]$ for the sequence



              $$ x_n = f(x_{n-1})$$






              share|cite|improve this answer









              $endgroup$



              This can be proved using the Banach Fixed Point Theorem.



              Intuitively, the BFPT tells us that if there is some function $F$ such that the distance between any two points $x$ and $y$ (when scaled by a constant $q$) is always larger than the distance between the corresponding images ($F(x)$ and $F(y)$), then the sequence
              $$x_n = F(x_{n-1}) $$
              will converge to a unique fixed point. For a more rigorous treatment of the BFPT statement: https://en.wikipedia.org/wiki/Banach_fixed-point_theorem.



              Since in this case you know that
              $$|f'(x)| leq t $$
              This implies that



              $$Big|frac{f(x) - f(y)}{x - y}Big| leq t$$



              for all possible points $x,y in [a,b]$. Think about why this is the case (Hint: Use the mean value theorem).



              This implies that



              $$| f(x) - f(y)| < t |x-y| $$



              which is what the BFPT requires. From here, we can just apply the BFPT to state that there is a fixed point in $[a,b]$ for the sequence



              $$ x_n = f(x_{n-1})$$







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered 4 hours ago









              Sean LeeSean Lee

              483211




              483211












              • $begingroup$
                Should probably expressly mention that the space $[0, 1]$ is complete, since that is a necessary condition for the Banach fixed point theorem to hold. For an easy counterexample demonstrating that you need completeness, consider $f(x) = x/2$ on $(0, 1]$.
                $endgroup$
                – Robert Shore
                4 hours ago










              • $begingroup$
                @RobertShore Yes, you are right, thanks!
                $endgroup$
                – Sean Lee
                4 hours ago


















              • $begingroup$
                Should probably expressly mention that the space $[0, 1]$ is complete, since that is a necessary condition for the Banach fixed point theorem to hold. For an easy counterexample demonstrating that you need completeness, consider $f(x) = x/2$ on $(0, 1]$.
                $endgroup$
                – Robert Shore
                4 hours ago










              • $begingroup$
                @RobertShore Yes, you are right, thanks!
                $endgroup$
                – Sean Lee
                4 hours ago
















              $begingroup$
              Should probably expressly mention that the space $[0, 1]$ is complete, since that is a necessary condition for the Banach fixed point theorem to hold. For an easy counterexample demonstrating that you need completeness, consider $f(x) = x/2$ on $(0, 1]$.
              $endgroup$
              – Robert Shore
              4 hours ago




              $begingroup$
              Should probably expressly mention that the space $[0, 1]$ is complete, since that is a necessary condition for the Banach fixed point theorem to hold. For an easy counterexample demonstrating that you need completeness, consider $f(x) = x/2$ on $(0, 1]$.
              $endgroup$
              – Robert Shore
              4 hours ago












              $begingroup$
              @RobertShore Yes, you are right, thanks!
              $endgroup$
              – Sean Lee
              4 hours ago




              $begingroup$
              @RobertShore Yes, you are right, thanks!
              $endgroup$
              – Sean Lee
              4 hours ago











              1












              $begingroup$

              To prove this from scratch note that by MVT $$|x_n-x_m| leq |x_{m+1}-x_m|+|x_{m+2}-x_{m+1}|$$ $$+cdots+|x_{n}-x_{n-1}|leq |x_{m+1}-x_m| (1+t+t^{2}+cdots+t^{n+m-1})$$ for $n >m$. Also $|x_{m+1}-x_m| leq t^{m-1} |x_2-x_1|$. Using the convergence of the geometric series $sum t^{n}$ conclude that ${x_n}$ is Cauchy. If $x =lim x_n$ then the definition of $x_n$'s and continuity of $f$ tells you that $f(x)=x$. Uniqueness follows easily by MVT.






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                To prove this from scratch note that by MVT $$|x_n-x_m| leq |x_{m+1}-x_m|+|x_{m+2}-x_{m+1}|$$ $$+cdots+|x_{n}-x_{n-1}|leq |x_{m+1}-x_m| (1+t+t^{2}+cdots+t^{n+m-1})$$ for $n >m$. Also $|x_{m+1}-x_m| leq t^{m-1} |x_2-x_1|$. Using the convergence of the geometric series $sum t^{n}$ conclude that ${x_n}$ is Cauchy. If $x =lim x_n$ then the definition of $x_n$'s and continuity of $f$ tells you that $f(x)=x$. Uniqueness follows easily by MVT.






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  To prove this from scratch note that by MVT $$|x_n-x_m| leq |x_{m+1}-x_m|+|x_{m+2}-x_{m+1}|$$ $$+cdots+|x_{n}-x_{n-1}|leq |x_{m+1}-x_m| (1+t+t^{2}+cdots+t^{n+m-1})$$ for $n >m$. Also $|x_{m+1}-x_m| leq t^{m-1} |x_2-x_1|$. Using the convergence of the geometric series $sum t^{n}$ conclude that ${x_n}$ is Cauchy. If $x =lim x_n$ then the definition of $x_n$'s and continuity of $f$ tells you that $f(x)=x$. Uniqueness follows easily by MVT.






                  share|cite|improve this answer









                  $endgroup$



                  To prove this from scratch note that by MVT $$|x_n-x_m| leq |x_{m+1}-x_m|+|x_{m+2}-x_{m+1}|$$ $$+cdots+|x_{n}-x_{n-1}|leq |x_{m+1}-x_m| (1+t+t^{2}+cdots+t^{n+m-1})$$ for $n >m$. Also $|x_{m+1}-x_m| leq t^{m-1} |x_2-x_1|$. Using the convergence of the geometric series $sum t^{n}$ conclude that ${x_n}$ is Cauchy. If $x =lim x_n$ then the definition of $x_n$'s and continuity of $f$ tells you that $f(x)=x$. Uniqueness follows easily by MVT.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 3 hours ago









                  Kavi Rama MurthyKavi Rama Murthy

                  65k42766




                  65k42766






















                      Grace is a new contributor. Be nice, and check out our Code of Conduct.










                      draft saved

                      draft discarded


















                      Grace is a new contributor. Be nice, and check out our Code of Conduct.













                      Grace is a new contributor. Be nice, and check out our Code of Conduct.












                      Grace is a new contributor. Be nice, and check out our Code of Conduct.
















                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3138183%2fconvergence-to-a-fixed-point%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      Popular posts from this blog

                      Knooppunt Holsloot

                      Altaar (religie)

                      Gregoriusmis