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If $\lim\limits_{x\to 0}f(x)=0$ and $X_n=(-1)^n/n$ then $\lim\limits_{n\to\infty}f(X_n)=0$

How do I prove it for every sequence $X_n$?

Thanks.

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For every sequence? Is your function continuous? –  Sigur Dec 21 '12 at 22:42
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Perhaps you could try showing that $X_n \to 0$? –  copper.hat Dec 21 '12 at 22:43

1 Answer 1

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The usual definition of a limit $\lim_{x\rightarrow a} f(x)$ is that $$\lim_{x\rightarrow a} f(x) = L$$ if and only if $$\lim_{n\rightarrow\infty}f(a_n) = L$$ for each sequence $(a_n)_{n\in\mathbb{N}}$ with $\lim_{n\rightarrow\infty}a_n = a$.

As $\lim_{n\rightarrow \infty} X_n = 0$, you get $$ \lim_{n\rightarrow \infty} f(X_n) = 0$$ by definition.

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