# sequence with convergent subsequence

• Sep 24th 2011, 03:31 PM
wopashui
sequence with convergent subsequence
If $(a_{n_k})$ is a convergent subsequence of $(a_n)$, show that lim inf $_{n--> infinity}$ $a_n$ $<=$ $lim_{k-->infinity}$ $a_{n_k}$ $<=$ lim sup $_{n-->infinity}$ $a_n$.

Im not sure what lim inf $_{n--> infinity}$ $a_n$ [tex] and lim sup $_{n-->infinity}$ $a_n$ mean, since $inf a_n$ and $sup a_n$ are a number
• Sep 24th 2011, 03:55 PM
Plato
Re: sequence with convergent subsequence
Quote:

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

This basically an easy question.
By definition of subsequence it must be the case that $k\le n_k$.
Thus $\{a_{n_k}:n_k\ge N\}\subseteq\{a_n:n\ge N\}$.
• Sep 25th 2011, 04:08 PM
wopashui
Re: sequence with convergent subsequence
Quote:

Originally Posted by Plato
This basically an easy question.
By definition of subsequence it must be the case that $k\le n_k$.
Thus $\{a_{n_k}:n_k\ge N\}\subseteq\{a_n:n\ge N\}$.

hmm, but we are talking about the limit here, not just K and $n_k$
• Sep 25th 2011, 04:23 PM
Plato
Re: sequence with convergent subsequence
Quote:

Originally Posted by wopashui
hmm, but we are talking about the limit here, not just K and $n_k$

Are you sure that you understand what $~\liminf~$ means?
• Sep 25th 2011, 05:14 PM
wopashui
Re: sequence with convergent subsequence
Quote:

Originally Posted by Plato
Are you sure that you understand what $~\liminf~$ means?

that is what im not sure about, I dunno what it means
• Sep 25th 2011, 06:03 PM
Plato
Re: sequence with convergent subsequence
Quote:

Originally Posted by wopashui
that is what im not sure about, I dunno what it means

$A\subseteq B$ then $\inf(B)\le\inf(A)$
and $\sup(A)\le\sup(B)~?$
Suppose that $(a_n)$ is a sequence.
$\limsup(a_n)=\lim _{n \to \infty } \sup \left\{ {a_k :k \geqslant n} \right\}$
we need to first show that $supa_n$ and $infa_n$ exist, do we?