Prove there exists a limit at infinity for an increasing and bounded function
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 Real Numbers, x M, f(x) f(M).
Fix > 0. Choose M Real Numbers such that |a-f(M)|< . Then, x M, |a-f(x)|< . 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)