Let $f: (M, d) \rightarrow (N, \rho)$ be uniformly continuous. Prove or disprove that if M is complete, then $f(M)$ is complete.

If I am asking a previously posted question, please accept my apologies and tell me to bugger off. I saw a similar problem but the solution was dealing with a Bi-Lipschitz function or some such business.

I believe this statement to be true and here is a rough sketch of my reasoning:

Since $f$ is uniformly continuous, then $f$ maps Cauchy to Cauchy. Let $(x_n)$ be a Cauchy sequence in $M$. Since $M$ is complete, $x_n \rightarrow x \in M$. Again, because of $f$'s uniform continuity, we now have $(f(x_n))$ is Cauchy in $N$ and $f(x_n) \rightarrow f(x) \in N$. Thus $N$ is complete.

By the way, I am studying for an exam. This is certainly not homework. I gladly accept your criticisms. Thank you in advance for your help.

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    $\begingroup$ You should start with a Cauchy sequence on $f(M)$ and try to prove that it converges. You did the opposite. You took a Cauchy sequence on $M$. $\endgroup$ – Sigur May 15 '13 at 22:37
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    $\begingroup$ I think $f(M)$ is not necessarily complete. because $\Bbb R$ can be homeomorphically embedded in unit circle. the image is open and so not complete. just a raw idea! $\endgroup$ – user59671 May 15 '13 at 22:48
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    $\begingroup$ Let $d$ be the discrete metric on $M=(0,1)$ and $\rho$ be the standard metric on $N=M$. Consider the identity map from $M$ to $N$. $\endgroup$ – David Mitra May 15 '13 at 23:05
  • $\begingroup$ @DavidMitra. I think this even demonstrates Lipschitz continuity doesn't, on its own, preserve completeness. $\endgroup$ – Tom Collinge Jan 22 '17 at 14:31

Take $f:\mathbb R\longrightarrow \mathbb R $, $f(x)=\arctan x$. Then $$f(\mathbb R)=(-\frac\pi2,\frac\pi2)$$ and $f'(x)=\dfrac{1}{1+x^2}\leq 1$ which implies that $f$ is uniformly continuous.

However, $\mathbb R$ is complete, while $f(\mathbb R)=(-\frac\pi2,\frac\pi2)$ is not complete

  • $\begingroup$ Yes, thank you all. The hints and solution provide much clarity for me as I continue to study. Have a good day. $\endgroup$ – user78000 May 16 '13 at 15:41

Let $M:=\Bbb R$ and $N:=(-\pi/2,\pi/2)$ and $f:=\arctan$. This is uniformly continuous (and, also a homeomorphism), but $N$ is not complete.


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