Let a and b be numbers with a < b. Suppose

that the function $\displaystyle f : (a, b) \rightarrow R $ is monotonically increasing and bounded. Prove that $\displaystyle \lim

_{x \rightarrow a }

f(x) $ exists.

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

Idea 1:

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

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

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

The problem is, do I know that $\displaystyle \{ f(a_n) \} $ is monotone? Since $\displaystyle \{ 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, $\displaystyle 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.