- #1
kathrynag
- 598
- 0
Homework Statement
let f: [a,b] ---> R be monotone. Prove that f has a limit at a and b.
Homework Equations
The Attempt at a Solution
here is my proof:
We need to show that f has a limit at a and b.
Consider f:[a,b]--> R where f is increasing. Now for all x elements of [a,b] f(a)<f(x)<f(b), hence f is bounded and by the monotonocity, f cannot be oscillatory. Suppose x1<x2<...xk. Then f(a)<f(x1)<f(x2)...<f(xk)<f(b). So limx-->xk f(x)=f(xk)<f(b)and f has a limit at b. Then Limx-->x1 f(x)=f(x1)>f(a) and f has a limit at a.
I'm not so sure about this one...