Is the following statement is true /false?

Question

Let $X$ and $Y$ be metric spaces and let $f : X \to Y$ be a mapping.

Is the following statement true or false ?

If $Y$ is complete and if $f$ is continuous, then the image of every Cauchy sequence in $X$ is a Cauchy sequence in $Y$.

I was taking $f:[0,1]\to[0,1]$ I take $f(x) = x$ then this statement is true.

Am i correct or not? I'm not getting another counter examples.

Possible duplicate of Continuity, uniform continuity and preservation of Cauchy sequences in metric spaces. — J.-E. Pin, Jan 03 '18 at 09:11

score 2 · Accepted Answer · answered Jan 03 '18 at 08:56

2

Let $f:(0,1)\rightarrow{\bf{R}}$, $f(x)=1/x$. Consider the Cauchy sequence $(x_{n})\subseteq(0,1)$, $x_{n}=1/(n+1)$, the sequence $(f(x_{n}))$ is not Cauchy.

answered Jan 03 '18 at 08:56

user284331

56,315

thanks a lots @user84331 – Jan 03 '18 at 09:12

Is the following statement is true /false?

1 Answers1