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I am solving assignments in Real analysis but I am unable to think about how I can solve this question.

Let $f: ( 0, \infty ) \to \mathbb{R}$ be a continuous function. Does $f$ maps any Cauchy sequence to a Cauchy sequence.

I tried by taking $ {x_n} $ and using |$x_n $ - $x_m$| <$\epsilon$ for all $n, m > N$ ( $N$ is a natural number) . But how can I formulate the Cauchy sequence property of $x_n$ into $f(x_n$) ? I am unable to think about it.

Any hints please?

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take the map $x \mapsto \frac{1}{x}$ and the Cauchy sequence $\{\frac{1}{n}\}_{n \ge 1}$

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