Let f:[a,b]--->R be monotone. Prove that f has a limit at BOTH a and b.
Let's assume that $\displaystyle f$ is nondecreasing.
The set $\displaystyle S=\{f(x):x\in (a,b)\}$ is bounded below by $\displaystyle f(a)$.
So let $\displaystyle L=\inf(S)$. Show that $\displaystyle L$ is limit as $\displaystyle x\to a^+$.
Ok, so do I just show that using the limit definition or is there some other trick?
After that, I assume I should make T={f(x): x is in (a,b)} bounded above by f(b)
And then have L2=sup(T) and show L2 is the limit as x--->b-?