real analysis - Prove that if $f$ is eventually monotone and eventually bounded $Rightarrow lim_{xrightarrow infty} f(x)$ is finite

If the function $f$ is defined on an unbounded above domain $D \subseteq \Re $ and is eventually monotone and eventually bounded, then $ \lim_{x\rightarrow \infty} f(x)$ is finite

I tried to workout the proof as:

Since $f$ is eventually monotone $\Rightarrow \exists x^*, x^* \leq x_1 < x_2 $ we have $f(x_1) \leq f(x_2)$

and since $f$ is eventually bounded $\Rightarrow \exists \hat{x},\ \exists \ L \leq M \in \mathbb{R} \ s.t. \lim_{x\rightarrow \infty} f(x) = L \\\forall \ \hat{x}\leq x $

Take $x = max(\hat{x}, x^*)$ and we have $\lim_{x\rightarrow \infty} f(x) = L$