Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
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 1

up vote 1 down vote accepted

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 $B$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.