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Math Help - Proof lim 1/a_n=inf

  1. #1
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    Proof lim 1/a_n=inf

    Hellou

    I'm havin real troubles with this one:

    Proof that if lim_{n->\infty} a_n=0 then lim_{n->\infty} 1/a_n=+\infty<br />

    can anyone help me?

    thx
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  2. #2
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    Quote Originally Posted by hiddy View Post
    Hellou

    I'm havin real troubles with this one:

    Proof that if lim_{n->\infty} a_n=0 then lim_{n->\infty} 1/a_n=+\infty<br />

    can anyone help me?

    thx
    We must assume, of course, that \{a_n\} is either positive or positive almost always, otherwise the result is false (for example, \frac{-1}{n}\xrightarrow [n\to \infty]{}0 and  -n\xrightarrow [n\to \infty]{}-\infty , or take any sequence which converges to zero and has infinite number of zeroes among its elements... )

    Let us take any 0<R\in\mathbb{R} (we can assume positiviness since the sequence is positive).
    No , since a_n\xrightarrow [n\to \infty]{}0 we get that for \epsilon=\frac{1}{R} there exists  N_\epsilon\in\mathbb{N}\,\,\,s.t.\,\,\,n>N_\epsilo  n\Longrightarrow a_n<\frac{1}{R}  \Longrightarrow \frac{1}{a_n}>R\,\,\,\forall\,n>N_\epsilon and this means, according to the definition, that \frac{1}{a_n}\xrightarrow [n\to \infty] {}\infty

    Tonio
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  3. #3
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    thx!!!

    ps:
    i forgot to mention that all a_n are positive real numbers!
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