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.

Idea 1:

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.

Idea 2:

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.