"If $f$ is a continuous function in metric X; $f: X →R$, and prove or give a counter example with proof, for every Cauchy $a_n$ in X, the sequence of $(f(a_n))$ is convergent in R."
I need hand in this question. I proved this for the case of uniform continuous. But I could not solve this part. My attempt was giving counter example with $f(x)= 1/x$. But I could not construct it with Cauchy sequence. Could you help me on that?
Let $X$ be the open interval $(-\pi/2,\pi/2)$, with the metric inherited from the usual metric on $\mathbb{R}$.
Let $f:X\to \mathbb{R}$ be given by $f(x)=\tan(x)$.
Let $(x_n)$ be a sequence of elements of $X$ which converges in $\mathbb{R}$ to $\pi/2$.
Then $f$ is continuous, and the sequence $(x_n)$ is a Cauchy sequence in $X$, but the sequence $f(x_n)$ is unbounded, so doesn't converge in $\mathbb{R}$.