Here's what I have to prove: Let $f$ be defined on the interval $(a,b)$ and suppose that $f$ is increasing and bounded on $(a,b)$. Then prove that $\lim_{x\to a^{+}} f(x)$ exists.
To prove $\lim_{x\to a^{+}} f(x)$ exists, I defined $L:=\inf \{ f(x): x\in (a,b)\}$ and claimed that $\lim_{x\to a^{+}} f(x) = L$. Let $\epsilon >0$ be given. Then there exists $x' \in (a,b)$ such that $L\le f(x') < L+\epsilon $. Now, let $\delta = x' -a$. If $a<x<a+\delta = x'$ then $f(x) \le f(x') < L+ \epsilon $ i.e. $|f(x) - L| < \epsilon $.
Is my proof correct? Is there any better way to do it?