Prove there exists a limit at infinity for an increasing and bounded function

**hasbren** · April 24th 2011, 08:52 PM

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 $\in$ Real Numbers, $\forall$ x $\geqslant$ M, f(x) $\geqslant$ f(M).

Fix $\varepsilon$ > 0. Choose M $\in$ Real Numbers such that |a-f(M)|< $\varepsilon$ . Then, $\forall$ x $\geqslant$ M, |a-f(x)|< $\varepsilon$ . 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)

**chisigma** · April 25th 2011, 02:25 AM

A more general criterion of existence of finite limit for a sequence, no matter if it is 'neither increasing nor decreasing', is illustrated in...

http://www.mathhelpforum.com/math-he...ce-176658.html

Kind regards

$\chi$ $\sigma$

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