please I require help in showing that if $f$ is uniformly continuous on a bounded interval $(a,b)$, then $\lim_{x\to b^-} f(x)$ exists.
edit:
$f$ is uniformly continuous on $(a,b)$ implies that for every $\epsilon \gt 0$ there is a $\delta \gt 0 $ such that $|f(x) -f(y)| \lt \epsilon$ whenever $|x-y|\lt \delta$ for every $x,y$.
let $x_n$ and $y_n$ be sequences in $(a,b)$ that converge to $b$. Then there is a natural number $N$ such that for every $n \geq N$, $0