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Let $X,Y$ metric spaces. If $f:X\to{Y}$ is continuous, and $\{y_n\}$ is a Cauchy sequence in $Y$. Then, my question is $\{f^{-1}(y_n)\}$ is a Cauchy sequence in $X$?

I´m sorry, in a second thought, f is surjective and suppose X is complete.

Any suggestion or a counterexample.

Thank you all.

user126033
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  • Is your map surjective, or do you assume that each $y_n$ lies in the image? Otherwise the question doesn't make sense. 2) Have you thought about what happens when the sequence ${y_n}$ is constant? 3) If you assume that $X$ and $Y$ are complete, you can replace "Cauchy" with "convergent", which seems to make things easier to think about. Must the preimage of a convergent sequence under a continuous map be convergent?
    • – Pete L. Clark Feb 04 '14 at 05:07
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    For a fixed $n$, $f^{-1}(y_n)$ is a set of points. For instance, imagine $f: \mathbb{R} \to {p}$. There is no "sequence" in $\mathbb{R}$. – Braindead Feb 04 '14 at 05:36
  • Perhaps better to say that $\Bbb R$ is not a sequence. – Cameron Buie Feb 04 '14 at 05:46
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    No: let $f(x)=1/x; \text {for} ; x>0$ and $;y_n=1/n$. – Tony Piccolo Feb 04 '14 at 10:00