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Let $f:M\to N$ continuous, such that exist $c>0$ with $d(f(x),f(y))\geq cd(x,y)$ for all $x,y\in M$. Show that $f$ transform complete subspaces of $M$ in complete subspaces of $N$.

I know that an application uniformly continuous $f:M\to N$, transform the cauchy sequences $x_n$ in cauchy sequences $(f(x_n))$, perhaps this fact is important for the problem, any help pls! Regards!

Ahna Akbar
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Assuming it's a $\leq$, then $$d(f(x),f(y)) \leq cd(x,y),$$ so $f$ is Lipschitz and therefore uniformly continuous so it preserves Cauchy sequences.

Are you clear on how to show that uniformly continuous functions preserve Cauchy sequences?

Anthony Peter
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    Ok, thanks for the answer. First, If $f:M\to N$ is uniformly continuous, with $(x_n)$ a cauchy sequence, by definition of uniff. cont., given $\epsilon>0$, we can find some $\delta>0$ s.t., for all $x,y\in M$, $d(x,y)<\delta\implies d(f(x),f(y))<\epsilon$. From $\delta$, we can find some $n_0\in\mathbb{N}$ s.t. for all $m,n>n_0\implies d(x_n,x_m)<\delta\implies d(f(x_n),f(x_m))<\epsilon$, and we are done. Second, in my book, say $\geq$, so my question. And third, how ends this?? regards! – Ahna Akbar Sep 25 '15 at 02:06
  • @AhnaAkbarperez Absolutely – Anthony Peter Sep 25 '15 at 02:07
  • sorry, I edited my answer. – Ahna Akbar Sep 25 '15 at 02:10
  • @AhnaAkbarperez How ends what? – Anthony Peter Sep 25 '15 at 02:10