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

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    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 \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)
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  2. #2
    MHF Contributor chisigma's Avatar
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    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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