# Let X be a metric space and let $f : X\rightarrow R$ be a continuous function. Pick out the true statements

Let $X$ be a metric space and let $f : X\rightarrow R$ be a continuous function. Pick out the true statements. (a) $f$ always maps Cauchy sequences into Cauchy sequences. (b) If $X$ is compact, then $f$ always maps Cauchy sequences into Cauchy sequences. (c) If $X = R^n$, then $f$ always maps Cauchy sequences into Cauchy sequences.

If $f$ is uniformly continuous then it maps a cauchy sequence to a cauchy sequence. So, a is not true and b is true. What about c?

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This might be helpful: en.wikipedia.org/wiki/Cauchy-continuous_function –  Shahab Sep 1 '12 at 1:23
Note that you have not actually proved that (a) is false. You need to exhibit a counterexample. –  Brian M. Scott Sep 1 '12 at 2:30

## 1 Answer

Hint: If $(x_k)$ is Cauchy in $\Bbb R^n$, then it is bounded and thus contained in a closed ball $B$ of finite radius. Now note that $B$ is compact in $\Bbb R^n$, and consider the function $f$ restricted to $B$.

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