Let a and b be numbers with a < b. Suppose
that the function f : (a, b) \rightarrow R is monotonically increasing and bounded. Prove that \lim<br /> _{x \rightarrow a }<br /> f(x) exists.

I have two ideas, and faults in both, please check.

Idea 1:

Now, let the sequence \{ a_n \} be of (a,b) such that it converges to the point a.

Note that the sequence \{ f(a_n) \} is monotonically decreasing and bounded, so it converges to a limit point, say L.

So we then have \lim _{n \rightarrow \infty } f( a_n) = L

The problem is, do I know that \{ f(a_n) \} is monotone? Since \{ a_n \} 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, f(a_n) \rightarrow f(a) , so the limit exist. But f(a) is not defined by this function...

Am I getting close? Please help! Thanks.