Let a and b be numbers with a < b. Suppose
that the function is monotonically increasing and bounded. Prove that exists.
I have two ideas, and faults in both, please check.
Now, let the sequence be of (a,b) such that it converges to the point a.
Note that the sequence is monotonically decreasing and bounded, so it converges to a limit point, say L.
So we then have
The problem is, do I know that is monotone? Since could very well jump back and forth before it converges to a.
Since the function f is bounded and monotone, it is therefore continuous, then by definition of continuity, , so the limit exist. But f(a) is not defined by this function...
Am I getting close? Please help! Thanks.