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Math Help - Proving limits help please???

  1. #1
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    Proving limits help please???

    Hi, can anyone give me a hand with this question, i honestly dont know what to do!!

    Prove that lim (n tends to infinity) 1/an = 0 when an is a sequence with an > 0 for n being natural, and lim (n tends to infinity) an = infinity.
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  2. #2
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    Quote Originally Posted by Bexii View Post
    Hi, can anyone give me a hand with this question, i honestly dont know what to do!!

    Prove that lim (n tends to infinity) 1/an = 0 when an is a sequence with an > 0 for n being natural, and lim (n tends to infinity) an = infinity.
    1 divided by a very big number is a very small number.. so if you divide something by infinity, you will get infinite small number = 0.
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  3. #3
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    i was told i had to use definitions in my answer...
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  4. #4
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    \forall \,\epsilon >0,\,\exists \,k\in \mathbb{N}\mid n\ge k\implies \left| \frac1n\right|<\epsilon .

    But this is quite straightforward, so we get that \frac1n<\epsilon which means that n>\frac1\epsilon, so for any k\in\mathbb N being greater than \frac1\epsilon, the above condition is fulfilled. For example, it's enough to take k=\left\lfloor \frac{1}{\epsilon} \right\rfloor +1.
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