## Monotonic Convergence of a function

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 $\lim
_{x \rightarrow a }
f(x)$
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...