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Math Help - Question about converging sequences.

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    Question about converging sequences.

    I know that a converging sequence a(n) to an a∈R is bounded. (I am pretty sure this is proven.) Is the above also valid for sequences converging to 0, i.e. Nullsequences -- Zerosequencies?

    If yes, can this be proven somehow with an equality or any other way? If yes, how?

    thank you
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Question about converging sequences.

    Quote Originally Posted by nappysnake View Post
    I know that a converging sequence a(n) to an a∈R is bounded. (I am pretty sure this is proven.) Is the above also valid for sequences converging to 0, i.e. Nullsequences -- Zerosequencies?
    Of course is also valid. Hint Choose \epsilon =1 , if \lim_{n\to \infty}a_n=0 there exists n_0\in \mathbb{N} such that |a_n|<1 if n\geq n_0 that is, all terms of the sequence are in (-1,1) except for a finite number.
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    Re: Question about converging sequences.

    amazing, thank you soo much Fernando for the explanation!
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