Suppose $f$ is a uniformly continuous mapping of a metric space $X$ into a metric space $Y$. If {$x_n$} is a Cauchy sequence in $X$, then {$f(x_n)$} is a Cauchy sequence in $Y$.
Give an example to show that this statement is not true if $f$ is continuous, but not uniformly continuous.
I was thinking maybe something like $\frac{x}{n}$? Not really sure if I understand this problem correctly. Any help would be great. Thank you.
Consider $a_n = \frac{1}{n}$ This is clearly cauchy, as it converges. Now consider $f: \mathbb{R/\{0\}} \to \mathbb{R/\{0\}}$, $$f(x) = \frac{1}{x}$$