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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.

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  • $\begingroup$ 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. $\endgroup$ – xbh Dec 5 '18 at 3:37
  • $\begingroup$ @xbh we haven't studied that theorem, so on this homework that wouldn't be the answer my prof is looking for. $\endgroup$ – user610107 Dec 5 '18 at 3:39
  • $\begingroup$ 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. $\endgroup$ – xbh Dec 5 '18 at 3:47
  • $\begingroup$ The sup norm of $F_j-F_k$ for $j\geq k$ can be directly bounded above using a geometric series, showing that $\{F_k\}$ is uniformly Cauchy. Then use the completeness of $\mathcal{C}([-r,r])$. $\endgroup$ – carmichael561 Dec 5 '18 at 3:48
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$$F_k(x)=\sum_{n=1}^kx^n = \frac{x(1-x^k)}{1-x}$$ converges to $$F(x) = \frac{x}{1-x}$$

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

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

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  • $\begingroup$ 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 $f_n:X\to Y$ of continuous functions converges uniformly, with respect to the metrics $d,e$, to $f:X\to Y,$ then $f$ is continuous. One method of proof is to show that if $\lim_{j\to \infty}d(x_j,x)=0$ then $\lim_{j\to \infty}e(f(x_j),f(x))=0$.... But in this case we have an explicit formula for $F$, and it is clearly continuous anyway.....+1 $\endgroup$ – DanielWainfleet Dec 5 '18 at 10:30
  • $\begingroup$ @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 $\endgroup$ – Damien Dec 5 '18 at 10:37
  • $\begingroup$ 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 $f_n:X\to \Bbb R$ be a sequence of continuous functions converging pointwise to a $ continuous$ $f:X\to \Bbb R.$ If $\forall n \in \Bbb N\;\forall x\in X\; (f_{n+1}(x)\leq f_n(x)\,)$ then $f_n$ converges uniformly to $f$. $\endgroup$ – DanielWainfleet Dec 6 '18 at 0:08
  • $\begingroup$ @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 $\endgroup$ – Damien Dec 6 '18 at 8:43

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