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Math Help - Tail of a Sequence

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    Tail of a Sequence

    Prove a sequence S_n converges iff its tail S_n+M converges.
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    Senior Member abhishekkgp's Avatar
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    Re: Tail of a Sequence

    Quote Originally Posted by veronicak5678 View Post
    Prove a sequence S_n converges iff its tail S_n+M converges.
    do you mean S_n converges iff S_{n+M} converges?
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    Re: Tail of a Sequence

    Yes, that's what I mean. Sorry I don't know Latex!
    So... any ideas?
    Last edited by veronicak5678; August 13th 2011 at 04:53 PM.
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    Senior Member abhishekkgp's Avatar
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    Re: Tail of a Sequence

    Quote Originally Posted by veronicak5678 View Post
    Yes, that's what I mean. Sorry I don't know Latex!
    So... any ideas?
    you just have to use the definition of convergence.

    Assume S_n \rightarrow a. we will show that S_{n+M} \rightarrow a.

    S_n \rightarrow a gives us:
    \forall \epsilon >0, \exists N \text{ such that } n>N \Rightarrow |S_n-a|<\epsilon

    To prove S_{n+M} \rightarrow a, that is:

    \forall \epsilon>0, \exists N^{'} \text{ such that } n>N^{'} \Rightarrow |S_{n+M}-a|<\epsilon. We easily see that N^{'}=N-M does the job.

    Now can you try and do the 'only if' part.
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    Re: Tail of a Sequence

    I see what I was doing wrong. Thanks everyone.
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