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Let $f:X \rightarrow Y$ be a function. Prove that if $f$ is continuous, then for every convergent sequence $(x_n)$ $\displaystyle\lim_{n\rightarrow \infty}f(x_n)=f(\lim_{n \rightarrow \infty} x_n)$

My attempt: Assume $f$ is continuous. Then $\forall \epsilon >0, \exists \delta > 0$ such that $\forall p \in X$ and $q \in X: d(p,q) < \delta \implies d(f(q),f(q)) < \epsilon.$

And also assume that $(x_n)$ converges. Then $\exists L$ such that $\forall \epsilon >0, \exists N$ such that $n>N \implies d(x_n, L) < \epsilon.$

I need to use these definitions to show lim$_{n\rightarrow \infty}\,f(x_n)=f(\text{lim}_{n \rightarrow \infty} \, x_n)$ . Any hints?

Edit: lim$_{n \rightarrow \infty}(x_n) \in X$

Pedro
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    You need $\lim \limits_{n\to \infty}(x_n)\in X$ as an hypothesis, otherwise the statement doesn't make sense. – Git Gud Dec 01 '13 at 19:11

2 Answers2

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If you define $L:=\lim_{n \rightarrow \infty}x_n$, and you know that $L$ is in the domain of your function, then you have $|f(x_n) - f(L)| < \epsilon$ whenever $|x_n - L|$ is less than some appropriate $\delta$. But you know by hypothesis that you can find an $N$ where this is true for every $n>N$.

That's a pretty big hint, but I'll let you polish it into a proof.

Matt R.
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    How do you know that $L$ is in the domain of the function? thanks – GniruT Jan 28 '16 at 14:54
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    @GniruT: Note that the assumption $\lim_{n \rightarrow \infty} x_n \in X$ is a necessary assumption, and was added in an edit by the poster. – Matt R. Jan 31 '16 at 14:27
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I'm replying with a possible answer I need confirmation on:

Let $L:=\lim_{n \rightarrow \infty}x_n$, and let $L$ be in the domain of $f$. We want to prove the sequence ${f(x_n)}$ converges to $f(L)$, namely $\forall\epsilon >0 \ \exists N$ such that $|f(x_n)-f(L)|<\epsilon$ for $n>N$.

$f$ is continuous in $x=L$ which means that $\forall\epsilon >0 \ \exists\delta$ such that $\forall x_n\in Dom(f), 0<|x_n-L|<\delta \to |f(x_n)-f(L)|<\epsilon $

We know $\lim_{n \rightarrow \infty}x_n=L$ which means that for any given $\delta$ I can find an $N$ for which $|x_m-L|<\delta$ for $n>N$

This means that given any $\epsilon$ I can find $N$ such that $|f(x_n)-f(L)|<\epsilon$, as I wanted to prove.

In short, by giving a value for $\epsilon$, the continuity gives a value for $\delta$, and given a value for $\delta$ the fact that $\lim_{n \rightarrow \infty}x_n=L$ gives a value for N

Shomar
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