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If $\{h_{k}\}_{k=1}^{\infty}$ is a sequence of real-valued continuous functions on the real line, and $0 and all $k$. Assume that $h_{k}(x)\to 0$ uniformly on $\mathbb{R}$. Is it true that $h_{k}(x+x_{k})\to 0$ as $k\to \infty$, for any sequence of points $\{x_{k}\}\subset \mathbb{R} $ and any $x\in \mathbb{R}$? If so how to prove it?

Thanks for help in advance!

Ok, what I have tried: I'm trying to use induction on $x_{k}$: Fix any $x\in\mathbb{R}, $we have

for $x_{1}$: $h_{k}(x+x_{1})\to 0$

for $x_{2}$: $h_{k}(x+x_{2})\to 0$

for $x_{3}$: $h_{k}(x+x_{3})\to 0$

.

.

.

for $x_{m}$: $h_{k}(x+x_{m})\to 0$

so, its true for all $m$, hence for $k$. Not sure!

Edit: I had a typo, sorry. My previous question was about $h_{k}(x_{k})\to 0$, but the correct question is about $h_{k}(x+x_{k})\to 0$

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    @DejanGoc: I know, but lets assume it is attained at these points2012-06-18

2 Answers 2

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This should be fairly direct.

The statement that $h_k(x) \to 0$ uniformly on $\mathbb{R}$ means that for any $\epsilon > 0$, there exists some $K_\epsilon$ such that $k>K_\epsilon$ implies that $|h_k(x)|<\epsilon$ for all $x \in \mathbb{R}$. If $\{x_k\}$ is some sequence in $\mathbb{R}$, then we certainly have that $|h_k(x_k)| < \epsilon$ for all $k>K_\epsilon$, and so $h_k(x_k) \to 0$.

Uniform convergence is a very powerful property.

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    He changed the question, but your proof is still correct (with $x+x_k$ instead of $x_k$). That is how nice uniform convergence is =)2012-06-18
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The idea Greg Zitelli points out in his answer should work with the new version of the problem too.

I'd just like to add that your idea of proof (if I understand it correctly) won't work. Here's an example that should demonstrate what the problem is. Let $f_n(x)=\begin{cases}0;&x and let $x_n=n$. Then for every term $x_i$ of this sequence you have: $f_n(x_i)\to0$, but $f_n(x_n)\to1$. Of course here the convergence is not uniform and the functions are not continuous, but it still shows that the convergence of $f_n(x_n)$ does not follow from that of $f_n(x_i)$.