Show (F$_{k}$) converges uniformly to some continuous function












0














Suppose ${0<r<1}$. For each k $in$ $mathbb{N}$, define F$_{k}$ $in$ C$bigl($[-r,r]$bigr)$ by F$_{k}$(x) = $sum_{n=1}^k$ x$^{n}$.



Show (F$_{k}$) converges uniformly to some continuous function f $in$ C$bigl($[-r,r]$bigr)$.



Somehow this has something to do with compact sets and metric spaces, as that was the section this homework was posted in. I'm not sure how they relate, however.






real-analysis convergence metric-spaces compactness riemann-sum







share|cite|improve this question
























  • I suppose you should use the Dini theorem, by first calculating the limit function and verify its continuity, then Dini theorem gets you there. Dini theorem is related to the compactness of domain.
    – xbh
    Dec 5 '18 at 3:37










  • @xbh we haven't studied that theorem, so on this homework that wouldn't be the answer my prof is looking for.
    – kendal
    Dec 5 '18 at 3:39










  • Then first compute the limit function, then prove the uniform convergence by definition. If that's the case, you might not use the compactness at all.
    – xbh
    Dec 5 '18 at 3:47










  • The sup norm of $F_j-F_k$ for $jgeq k$ can be directly bounded above using a geometric series, showing that ${F_k}$ is uniformly Cauchy. Then use the completeness of $mathcal{C}([-r,r])$.
    – carmichael561
    Dec 5 '18 at 3:48


















0














Suppose ${0<r<1}$. For each k $in$ $mathbb{N}$, define F$_{k}$ $in$ C$bigl($[-r,r]$bigr)$ by F$_{k}$(x) = $sum_{n=1}^k$ x$^{n}$.



Show (F$_{k}$) converges uniformly to some continuous function f $in$ C$bigl($[-r,r]$bigr)$.



Somehow this has something to do with compact sets and metric spaces, as that was the section this homework was posted in. I'm not sure how they relate, however.






real-analysis convergence metric-spaces compactness riemann-sum







share|cite|improve this question
























  • I suppose you should use the Dini theorem, by first calculating the limit function and verify its continuity, then Dini theorem gets you there. Dini theorem is related to the compactness of domain.
    – xbh
    Dec 5 '18 at 3:37










  • @xbh we haven't studied that theorem, so on this homework that wouldn't be the answer my prof is looking for.
    – kendal
    Dec 5 '18 at 3:39










  • Then first compute the limit function, then prove the uniform convergence by definition. If that's the case, you might not use the compactness at all.
    – xbh
    Dec 5 '18 at 3:47










  • The sup norm of $F_j-F_k$ for $jgeq k$ can be directly bounded above using a geometric series, showing that ${F_k}$ is uniformly Cauchy. Then use the completeness of $mathcal{C}([-r,r])$.
    – carmichael561
    Dec 5 '18 at 3:48
















0












0








0







Suppose ${0<r<1}$. For each k $in$ $mathbb{N}$, define F$_{k}$ $in$ C$bigl($[-r,r]$bigr)$ by F$_{k}$(x) = $sum_{n=1}^k$ x$^{n}$.



Show (F$_{k}$) converges uniformly to some continuous function f $in$ C$bigl($[-r,r]$bigr)$.



Somehow this has something to do with compact sets and metric spaces, as that was the section this homework was posted in. I'm not sure how they relate, however.






real-analysis convergence metric-spaces compactness riemann-sum







share|cite|improve this question















Suppose ${0<r<1}$. For each k $in$ $mathbb{N}$, define F$_{k}$ $in$ C$bigl($[-r,r]$bigr)$ by F$_{k}$(x) = $sum_{n=1}^k$ x$^{n}$.



Show (F$_{k}$) converges uniformly to some continuous function f $in$ C$bigl($[-r,r]$bigr)$.



Somehow this has something to do with compact sets and metric spaces, as that was the section this homework was posted in. I'm not sure how they relate, however.






real-analysis convergence metric-spaces compactness riemann-sum




real-analysis convergence metric-spaces compactness riemann-sum






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 7 '18 at 18:35









Math Avengers

709




709










asked Dec 5 '18 at 3:33









kendalkendal

337




337












  • I suppose you should use the Dini theorem, by first calculating the limit function and verify its continuity, then Dini theorem gets you there. Dini theorem is related to the compactness of domain.
    – xbh
    Dec 5 '18 at 3:37










  • @xbh we haven't studied that theorem, so on this homework that wouldn't be the answer my prof is looking for.
    – kendal
    Dec 5 '18 at 3:39










  • Then first compute the limit function, then prove the uniform convergence by definition. If that's the case, you might not use the compactness at all.
    – xbh
    Dec 5 '18 at 3:47










  • The sup norm of $F_j-F_k$ for $jgeq k$ can be directly bounded above using a geometric series, showing that ${F_k}$ is uniformly Cauchy. Then use the completeness of $mathcal{C}([-r,r])$.
    – carmichael561
    Dec 5 '18 at 3:48




















  • I suppose you should use the Dini theorem, by first calculating the limit function and verify its continuity, then Dini theorem gets you there. Dini theorem is related to the compactness of domain.
    – xbh
    Dec 5 '18 at 3:37










  • @xbh we haven't studied that theorem, so on this homework that wouldn't be the answer my prof is looking for.
    – kendal
    Dec 5 '18 at 3:39










  • Then first compute the limit function, then prove the uniform convergence by definition. If that's the case, you might not use the compactness at all.
    – xbh
    Dec 5 '18 at 3:47










  • The sup norm of $F_j-F_k$ for $jgeq k$ can be directly bounded above using a geometric series, showing that ${F_k}$ is uniformly Cauchy. Then use the completeness of $mathcal{C}([-r,r])$.
    – carmichael561
    Dec 5 '18 at 3:48


















I suppose you should use the Dini theorem, by first calculating the limit function and verify its continuity, then Dini theorem gets you there. Dini theorem is related to the compactness of domain.
– xbh
Dec 5 '18 at 3:37




I suppose you should use the Dini theorem, by first calculating the limit function and verify its continuity, then Dini theorem gets you there. Dini theorem is related to the compactness of domain.
– xbh
Dec 5 '18 at 3:37












@xbh we haven't studied that theorem, so on this homework that wouldn't be the answer my prof is looking for.
– kendal
Dec 5 '18 at 3:39




@xbh we haven't studied that theorem, so on this homework that wouldn't be the answer my prof is looking for.
– kendal
Dec 5 '18 at 3:39












Then first compute the limit function, then prove the uniform convergence by definition. If that's the case, you might not use the compactness at all.
– xbh
Dec 5 '18 at 3:47




Then first compute the limit function, then prove the uniform convergence by definition. If that's the case, you might not use the compactness at all.
– xbh
Dec 5 '18 at 3:47












The sup norm of $F_j-F_k$ for $jgeq k$ can be directly bounded above using a geometric series, showing that ${F_k}$ is uniformly Cauchy. Then use the completeness of $mathcal{C}([-r,r])$.
– carmichael561
Dec 5 '18 at 3:48






The sup norm of $F_j-F_k$ for $jgeq k$ can be directly bounded above using a geometric series, showing that ${F_k}$ is uniformly Cauchy. Then use the completeness of $mathcal{C}([-r,r])$.
– carmichael561
Dec 5 '18 at 3:48












1 Answer
1






active

oldest

votes


















3














$$F_k(x)=sum_{n=1}^kx^n = frac{x(1-x^k)}{1-x}$$
converges to
$$F(x) = frac{x}{1-x}$$



Then
$$left|F_k(x)-F(x)right| = left|sum_{n=k+1}^infty x^nright| = |x^k F(x) | le r^k F(r)$$



The last value tends to $0$ when $k to infty$ and is independent of $x$. Uniform convergence follows.






share|cite|improve this answer





















  • The limit function $F$ is continuous by a widely-used (& not difficult) result that if $(X,d)$ and $(Y,e)$ are metric spaces and a sequence $f_n:Xto Y$ of continuous functions converges uniformly, with respect to the metrics $d,e$, to $f:Xto Y,$ then $f$ is continuous. One method of proof is to show that if $lim_{jto infty}d(x_j,x)=0$ then $lim_{jto infty}e(f(x_j),f(x))=0$.... But in this case we have an explicit formula for $F$, and it is clearly continuous anyway.....+1
    – DanielWainfleet
    Dec 5 '18 at 10:30












  • @DanielWainfleet Thank you for the information. I am an engineer and basically I am doing maths as an engineer ... My maths lessons are very far
    – Damien
    Dec 5 '18 at 10:37










  • Another result, called The Dini Theorem (perhaps to distinguish it from some other important results, especially in Measure Theory, by Dini): Let $X$ be a compact Hausdorff space (e.g. a closed bounded subset of $Bbb R$ ) and let $f_n:Xto Bbb R$ be a sequence of continuous functions converging pointwise to a $ continuous$ $f:Xto Bbb R.$ If $forall n in Bbb N;forall xin X; (f_{n+1}(x)leq f_n(x),)$ then $f_n$ converges uniformly to $f$.
    – DanielWainfleet
    Dec 6 '18 at 0:08












  • @DanielWainfleet Thank you again. In my answer, in particular, I used the fact that F() is bounded, which is related to the conditions of Dini Theorem. But I also used the particular characteristics of the series
    – Damien
    Dec 6 '18 at 8:43











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1 Answer
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active

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









3














$$F_k(x)=sum_{n=1}^kx^n = frac{x(1-x^k)}{1-x}$$
converges to
$$F(x) = frac{x}{1-x}$$



Then
$$left|F_k(x)-F(x)right| = left|sum_{n=k+1}^infty x^nright| = |x^k F(x) | le r^k F(r)$$



The last value tends to $0$ when $k to infty$ and is independent of $x$. Uniform convergence follows.






share|cite|improve this answer





















  • The limit function $F$ is continuous by a widely-used (& not difficult) result that if $(X,d)$ and $(Y,e)$ are metric spaces and a sequence $f_n:Xto Y$ of continuous functions converges uniformly, with respect to the metrics $d,e$, to $f:Xto Y,$ then $f$ is continuous. One method of proof is to show that if $lim_{jto infty}d(x_j,x)=0$ then $lim_{jto infty}e(f(x_j),f(x))=0$.... But in this case we have an explicit formula for $F$, and it is clearly continuous anyway.....+1
    – DanielWainfleet
    Dec 5 '18 at 10:30












  • @DanielWainfleet Thank you for the information. I am an engineer and basically I am doing maths as an engineer ... My maths lessons are very far
    – Damien
    Dec 5 '18 at 10:37










  • Another result, called The Dini Theorem (perhaps to distinguish it from some other important results, especially in Measure Theory, by Dini): Let $X$ be a compact Hausdorff space (e.g. a closed bounded subset of $Bbb R$ ) and let $f_n:Xto Bbb R$ be a sequence of continuous functions converging pointwise to a $ continuous$ $f:Xto Bbb R.$ If $forall n in Bbb N;forall xin X; (f_{n+1}(x)leq f_n(x),)$ then $f_n$ converges uniformly to $f$.
    – DanielWainfleet
    Dec 6 '18 at 0:08












  • @DanielWainfleet Thank you again. In my answer, in particular, I used the fact that F() is bounded, which is related to the conditions of Dini Theorem. But I also used the particular characteristics of the series
    – Damien
    Dec 6 '18 at 8:43
















3














$$F_k(x)=sum_{n=1}^kx^n = frac{x(1-x^k)}{1-x}$$
converges to
$$F(x) = frac{x}{1-x}$$



Then
$$left|F_k(x)-F(x)right| = left|sum_{n=k+1}^infty x^nright| = |x^k F(x) | le r^k F(r)$$



The last value tends to $0$ when $k to infty$ and is independent of $x$. Uniform convergence follows.






share|cite|improve this answer





















  • The limit function $F$ is continuous by a widely-used (& not difficult) result that if $(X,d)$ and $(Y,e)$ are metric spaces and a sequence $f_n:Xto Y$ of continuous functions converges uniformly, with respect to the metrics $d,e$, to $f:Xto Y,$ then $f$ is continuous. One method of proof is to show that if $lim_{jto infty}d(x_j,x)=0$ then $lim_{jto infty}e(f(x_j),f(x))=0$.... But in this case we have an explicit formula for $F$, and it is clearly continuous anyway.....+1
    – DanielWainfleet
    Dec 5 '18 at 10:30












  • @DanielWainfleet Thank you for the information. I am an engineer and basically I am doing maths as an engineer ... My maths lessons are very far
    – Damien
    Dec 5 '18 at 10:37










  • Another result, called The Dini Theorem (perhaps to distinguish it from some other important results, especially in Measure Theory, by Dini): Let $X$ be a compact Hausdorff space (e.g. a closed bounded subset of $Bbb R$ ) and let $f_n:Xto Bbb R$ be a sequence of continuous functions converging pointwise to a $ continuous$ $f:Xto Bbb R.$ If $forall n in Bbb N;forall xin X; (f_{n+1}(x)leq f_n(x),)$ then $f_n$ converges uniformly to $f$.
    – DanielWainfleet
    Dec 6 '18 at 0:08












  • @DanielWainfleet Thank you again. In my answer, in particular, I used the fact that F() is bounded, which is related to the conditions of Dini Theorem. But I also used the particular characteristics of the series
    – Damien
    Dec 6 '18 at 8:43














3












3








3






$$F_k(x)=sum_{n=1}^kx^n = frac{x(1-x^k)}{1-x}$$
converges to
$$F(x) = frac{x}{1-x}$$



Then
$$left|F_k(x)-F(x)right| = left|sum_{n=k+1}^infty x^nright| = |x^k F(x) | le r^k F(r)$$



The last value tends to $0$ when $k to infty$ and is independent of $x$. Uniform convergence follows.






share|cite|improve this answer












$$F_k(x)=sum_{n=1}^kx^n = frac{x(1-x^k)}{1-x}$$
converges to
$$F(x) = frac{x}{1-x}$$



Then
$$left|F_k(x)-F(x)right| = left|sum_{n=k+1}^infty x^nright| = |x^k F(x) | le r^k F(r)$$



The last value tends to $0$ when $k to infty$ and is independent of $x$. Uniform convergence follows.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 5 '18 at 8:54









DamienDamien

52714




52714












  • The limit function $F$ is continuous by a widely-used (& not difficult) result that if $(X,d)$ and $(Y,e)$ are metric spaces and a sequence $f_n:Xto Y$ of continuous functions converges uniformly, with respect to the metrics $d,e$, to $f:Xto Y,$ then $f$ is continuous. One method of proof is to show that if $lim_{jto infty}d(x_j,x)=0$ then $lim_{jto infty}e(f(x_j),f(x))=0$.... But in this case we have an explicit formula for $F$, and it is clearly continuous anyway.....+1
    – DanielWainfleet
    Dec 5 '18 at 10:30












  • @DanielWainfleet Thank you for the information. I am an engineer and basically I am doing maths as an engineer ... My maths lessons are very far
    – Damien
    Dec 5 '18 at 10:37










  • Another result, called The Dini Theorem (perhaps to distinguish it from some other important results, especially in Measure Theory, by Dini): Let $X$ be a compact Hausdorff space (e.g. a closed bounded subset of $Bbb R$ ) and let $f_n:Xto Bbb R$ be a sequence of continuous functions converging pointwise to a $ continuous$ $f:Xto Bbb R.$ If $forall n in Bbb N;forall xin X; (f_{n+1}(x)leq f_n(x),)$ then $f_n$ converges uniformly to $f$.
    – DanielWainfleet
    Dec 6 '18 at 0:08












  • @DanielWainfleet Thank you again. In my answer, in particular, I used the fact that F() is bounded, which is related to the conditions of Dini Theorem. But I also used the particular characteristics of the series
    – Damien
    Dec 6 '18 at 8:43


















  • The limit function $F$ is continuous by a widely-used (& not difficult) result that if $(X,d)$ and $(Y,e)$ are metric spaces and a sequence $f_n:Xto Y$ of continuous functions converges uniformly, with respect to the metrics $d,e$, to $f:Xto Y,$ then $f$ is continuous. One method of proof is to show that if $lim_{jto infty}d(x_j,x)=0$ then $lim_{jto infty}e(f(x_j),f(x))=0$.... But in this case we have an explicit formula for $F$, and it is clearly continuous anyway.....+1
    – DanielWainfleet
    Dec 5 '18 at 10:30












  • @DanielWainfleet Thank you for the information. I am an engineer and basically I am doing maths as an engineer ... My maths lessons are very far
    – Damien
    Dec 5 '18 at 10:37










  • Another result, called The Dini Theorem (perhaps to distinguish it from some other important results, especially in Measure Theory, by Dini): Let $X$ be a compact Hausdorff space (e.g. a closed bounded subset of $Bbb R$ ) and let $f_n:Xto Bbb R$ be a sequence of continuous functions converging pointwise to a $ continuous$ $f:Xto Bbb R.$ If $forall n in Bbb N;forall xin X; (f_{n+1}(x)leq f_n(x),)$ then $f_n$ converges uniformly to $f$.
    – DanielWainfleet
    Dec 6 '18 at 0:08












  • @DanielWainfleet Thank you again. In my answer, in particular, I used the fact that F() is bounded, which is related to the conditions of Dini Theorem. But I also used the particular characteristics of the series
    – Damien
    Dec 6 '18 at 8:43
















The limit function $F$ is continuous by a widely-used (& not difficult) result that if $(X,d)$ and $(Y,e)$ are metric spaces and a sequence $f_n:Xto Y$ of continuous functions converges uniformly, with respect to the metrics $d,e$, to $f:Xto Y,$ then $f$ is continuous. One method of proof is to show that if $lim_{jto infty}d(x_j,x)=0$ then $lim_{jto infty}e(f(x_j),f(x))=0$.... But in this case we have an explicit formula for $F$, and it is clearly continuous anyway.....+1
– DanielWainfleet
Dec 5 '18 at 10:30






The limit function $F$ is continuous by a widely-used (& not difficult) result that if $(X,d)$ and $(Y,e)$ are metric spaces and a sequence $f_n:Xto Y$ of continuous functions converges uniformly, with respect to the metrics $d,e$, to $f:Xto Y,$ then $f$ is continuous. One method of proof is to show that if $lim_{jto infty}d(x_j,x)=0$ then $lim_{jto infty}e(f(x_j),f(x))=0$.... But in this case we have an explicit formula for $F$, and it is clearly continuous anyway.....+1
– DanielWainfleet
Dec 5 '18 at 10:30














@DanielWainfleet Thank you for the information. I am an engineer and basically I am doing maths as an engineer ... My maths lessons are very far
– Damien
Dec 5 '18 at 10:37




@DanielWainfleet Thank you for the information. I am an engineer and basically I am doing maths as an engineer ... My maths lessons are very far
– Damien
Dec 5 '18 at 10:37












Another result, called The Dini Theorem (perhaps to distinguish it from some other important results, especially in Measure Theory, by Dini): Let $X$ be a compact Hausdorff space (e.g. a closed bounded subset of $Bbb R$ ) and let $f_n:Xto Bbb R$ be a sequence of continuous functions converging pointwise to a $ continuous$ $f:Xto Bbb R.$ If $forall n in Bbb N;forall xin X; (f_{n+1}(x)leq f_n(x),)$ then $f_n$ converges uniformly to $f$.
– DanielWainfleet
Dec 6 '18 at 0:08






Another result, called The Dini Theorem (perhaps to distinguish it from some other important results, especially in Measure Theory, by Dini): Let $X$ be a compact Hausdorff space (e.g. a closed bounded subset of $Bbb R$ ) and let $f_n:Xto Bbb R$ be a sequence of continuous functions converging pointwise to a $ continuous$ $f:Xto Bbb R.$ If $forall n in Bbb N;forall xin X; (f_{n+1}(x)leq f_n(x),)$ then $f_n$ converges uniformly to $f$.
– DanielWainfleet
Dec 6 '18 at 0:08














@DanielWainfleet Thank you again. In my answer, in particular, I used the fact that F() is bounded, which is related to the conditions of Dini Theorem. But I also used the particular characteristics of the series
– Damien
Dec 6 '18 at 8:43




@DanielWainfleet Thank you again. In my answer, in particular, I used the fact that F() is bounded, which is related to the conditions of Dini Theorem. But I also used the particular characteristics of the series
– Damien
Dec 6 '18 at 8:43


















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Different font size/position of beamer's navigation symbols template's content depending on regular/plain...

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3 Let's suppose I'd like to divert the navigation symbols template (e.g. in order replace them by the frame numbers, see this bug report ;) I would be facing troubles regarding the font size/position, depending on regular/plain frames, as shown by the following MCE. How could I get rid of these troubles? documentclass{beamer} usetheme{PaloAlto} setbeamertemplate{navigation symbols}{insertframenumber/inserttotalframenumber} begin{document} begin{frame} frametitle{A regular frame} end{frame} begin{frame}[plain] frametitle{A plain frame} end{frame} end{document} beamer header-footer share | improve this question asked Dec 10 '18 a...
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