Let $E$ be a bounded subset of $\mathbb{R}$ and assume that the function $f:E\to\mathbb{R}$ takes Cauchy sequences in $E$ to Cauchy sequences in $\mathbb{R}$. Prove that $f$ is continuous on $E$ Is $f$ uniformly continuous on $E$?
For the first part I tried to show $\lim_{n\to\infty}x_{n}=x$ implies $\lim_{n\to\infty}f(x_{n})=f(x)$.To do this I used that $x_{n}$ is Cauchy in this case so $f(x_{n})$ is Cauchy so it converges to a number $L$, $|f(x)-f(x_{n})|<|f(x)-L|+|f(x_{n})-L|<\frac{\epsilon}{2}+|f(x)-L|$ after some $n \in\mathbb{N}$ why is $|f(x)-L|$ small? For the second part I think answer is no .