# Proof...monotonicity

• Oct 9th 2009, 11:32 AM
zhupolongjoe
Proof...monotonicity
Let f:[a,b]--->R be monotone. Prove that f has a limit at BOTH a and b.

• Oct 9th 2009, 12:45 PM
Plato
Quote:

Originally Posted by zhupolongjoe
Let f:[a,b]--->R be monotone. Prove that f has a limit at BOTH a and b.

Let's assume that $f$ is nondecreasing.
The set $S=\{f(x):x\in (a,b)\}$ is bounded below by $f(a)$.
So let $L=\inf(S)$. Show that $L$ is limit as $x\to a^+$.
• Oct 10th 2009, 11:10 AM
zhupolongjoe
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-?

Thanks