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Thread: sequence with convergent subsequence

  1. #1
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    sequence with convergent subsequence

    If $\displaystyle (a_{n_k})$ is a convergent subsequence of $\displaystyle (a_n)$, show that lim inf $\displaystyle _{n--> infinity}$ $\displaystyle a_n$ $\displaystyle <= $$\displaystyle lim_{k-->infinity}$$\displaystyle a_{n_k}$ $\displaystyle <=$ lim sup$\displaystyle _{n-->infinity}$$\displaystyle a_n$.

    Im not sure what lim inf $\displaystyle _{n--> infinity}$ $\displaystyle a_n$ [tex] and lim sup$\displaystyle _{n-->infinity}$$\displaystyle a_n$ mean, since $\displaystyle inf a_n$ and $\displaystyle sup a_n$ are a number
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  2. #2
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    Re: sequence with convergent subsequence

    Quote Originally Posted by wopashui View Post
    If $\displaystyle (a_{n_k})$ is a convergent subsequence of $\displaystyle (a_n)$, show that lim inf $\displaystyle _{n--> infinity}$ $\displaystyle a_n$ $\displaystyle <= $$\displaystyle lim_{k-->infinity}$$\displaystyle a_{n_k}$ $\displaystyle <=$ lim sup$\displaystyle _{n-->infinity}$$\displaystyle a_n$.
    This basically an easy question.
    By definition of subsequence it must be the case that $\displaystyle k\le n_k$.
    Thus $\displaystyle \{a_{n_k}:n_k\ge N\}\subseteq\{a_n:n\ge N\}$.
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    Re: sequence with convergent subsequence

    Quote Originally Posted by Plato View Post
    This basically an easy question.
    By definition of subsequence it must be the case that $\displaystyle k\le n_k$.
    Thus $\displaystyle \{a_{n_k}:n_k\ge N\}\subseteq\{a_n:n\ge N\}$.
    hmm, but we are talking about the limit here, not just K and $\displaystyle n_k$
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    Re: sequence with convergent subsequence

    Quote Originally Posted by wopashui View Post
    hmm, but we are talking about the limit here, not just K and $\displaystyle n_k$
    Are you sure that you understand what $\displaystyle ~\liminf~$ means?
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    Re: sequence with convergent subsequence

    Quote Originally Posted by Plato View Post
    Are you sure that you understand what $\displaystyle ~\liminf~$ means?
    that is what im not sure about, I dunno what it means
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    Re: sequence with convergent subsequence

    Quote Originally Posted by wopashui View Post
    that is what im not sure about, I dunno what it means
    Well then, no wonder you cannot answer questions about it.
    Does it make sense to you that if
    $\displaystyle A\subseteq B$ then $\displaystyle \inf(B)\le\inf(A)$
    and $\displaystyle \sup(A)\le\sup(B)~?$

    Suppose that $\displaystyle (a_n)$ is a sequence.
    $\displaystyle \limsup(a_n)=\lim _{n \to \infty } \sup \left\{ {a_k :k \geqslant n} \right\}$
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  7. #7
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    Re: sequence with convergent subsequence

    we need to first show that $\displaystyle supa_n$ and $\displaystyle infa_n$ exist, do we?
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