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Math Help - Simple question regarding sequences and subsequences

  1. #1
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    Simple question regarding sequences and subsequences

    I'm sure I'm overlooking or misunderstanding some obvious detail here, but nevertheless:

    It can be shown that for any sequence (a_n) that converges to some limit L, any subsequence (a_{nj}) of (a_n) also converges to L. Correct? But I don't quite understand this. Suppose we have the (arbitrary) sequence a_n = \{1, 2, 3, 4, 5 \}, which converges to 5. But then a_{nj} = \{1, 3 \} is a subsequence of (a_n), which converges to 3, no? What obvious detail am I missing here?

    Also, does every sequence necessarily have a subsequence?
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  2. #2
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    Quote Originally Posted by Ares_D1 View Post
    I'm sure I'm overlooking or misunderstanding some obvious detail here, but nevertheless:

    It can be shown that for any sequence (a_n) that converges to some limit L, any subsequence (a_{nj}) of (a_n) also converges to L. Correct? But I don't quite understand this. Suppose we have the (arbitrary) sequence a_n = \{1, 2, 3, 4, 5 \}, which converges to 5. But then a_{nj} = \{1, 3 \} is a subsequence of (a_n), which converges to 3, no? What obvious detail am I missing here?

    Also, does every sequence necessarily have a subsequence?
    Dear Ares_D1,

    If you consider the definition of the limit of a sequence, it is defined for infinite sequences only.( Limit of a sequence - Wikipedia, the free encyclopedia) Therefore for a finite sequence there is no such thing as a limit.
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