I would appreciate help improving this proof. Do I need to be more rigorous?

Cl: There exists a limit at infinity for an increasing and bounded function.

Pf: Suppose f is an increasing and bounded function. By the least upper bound property of real numbers, since f is bounded, there exists a sup(f), call it 'a'.

Since f is increasing, if M $\displaystyle \in$ Real Numbers, $\displaystyle \forall$ x $\displaystyle \geqslant$ M, f(x)$\displaystyle \geqslant$ f(M).

Fix $\displaystyle \varepsilon$ > 0. Choose M $\displaystyle \in$ Real Numbers such that |a-f(M)|< $\displaystyle \varepsilon$. Then, $\displaystyle \forall$ x$\displaystyle \geqslant$ M, |a-f(x)|< $\displaystyle \varepsilon$. Therefore we conclude, the limit as x approaches infinity of f = sup(f)=a.

(sorry for the messy tex stuff, I'm just a beginner)