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Thread: Convergent sequences

  1. #1
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    Convergent sequences

    1) Can you find a sequnce of real nos. {sn} which has no convergent subsequence and yet for which {|sn|} converges.

    I am not able to think of such a sequence.

    2) If a subsequence of {sn} of terms with even subscripts converges to L, as well as the subsequence with odd subscripts (converges to L i.e.) then, prove that the given sequence converges to L.

    Here I intuitively know the theorem statement is sort of obvious, but how do I go about the proof?

    Any help will be appreciated.
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  2. #2
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    1) Let $\displaystyle (s_n)$ be a sequence such that $\displaystyle (\vert s_n \vert )$ converges. Clearly $\displaystyle (s_n)$ must have an infinite number of positive (or negative) terms. Take $\displaystyle (s_{n_k})$ such that $\displaystyle 0 \leq s_{n_k}$ for all $\displaystyle n_k$ and $\displaystyle n_k < n_{k+1}$ then $\displaystyle (s_{n_k})$ is a subsequence of $\displaystyle (s_n)$, but $\displaystyle \vert s_n \vert$ converges.

    2)Use the fact that the odd and even numbers partition $\displaystyle \mathbb{N}$
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  3. #3
    Super Member Gamma's Avatar
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    more explicit for 2)

    I think for 2) more explicitly it is best to say:

    Given $\displaystyle \epsilon >0$, since the even and odd sequences converge to L, there should be an $\displaystyle N_{odd}$ and an $\displaystyle N_{even}$ such that for all odd n bigger than $\displaystyle N_{odd}$ you have $\displaystyle |x_n-L|<\epsilon$ and similarly for all even n bigger than $\displaystyle N_{even}$ you have $\displaystyle |x_n-L|<\epsilon$.

    Then let $\displaystyle N:=max\{N_{even},N_{odd}\}$ and you have for all $\displaystyle n>N$ that $\displaystyle |x_n-L|<\epsilon$ so the sequence converges to L as desired.
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  4. #4
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    For the first question I don't think it's possible. For if $\displaystyle s_n$ were bounded, it would have a convergent subsequence by Bolzano-Weierstrass. Thus the sequence must be unbounded. But this too isn't possible because if $\displaystyle |s_n|$ converges we have that by $\displaystyle -|s_n|\le s_n\le |s_n|$ (nothing is lost by assuming all terms positive really) that $\displaystyle s_n$ would converge. But $\displaystyle s_n$ is unbounded and thus diverges, and so must $\displaystyle |s_n|$.
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