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So we need to prove that the limit of any subsequence is sandwiched between the $\limsup$ and $\liminf$ of the original sequence.

The professor started off by showing that there is always a subsequence that converges to $\limsup a_n$, and also there is always a subsequence that converges to $\liminf a_n$. He also showed that $\limsup a_{n_{k}}$ is less than or equal to $\limsup a_n$. This part I also understand. But then how do you use these two facts to conclude that $\lim a_{n_{k}} \leq \limsup a_n$?

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  • $\begingroup$ Mode of attack: Assume $\lim a_{n_k} > \limsup a_n$ and try to obtain a contradiction. $\endgroup$ Oct 9 '15 at 20:59
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$\lim a_{n_k} \le \limsup a_{n_k}$!

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  • $\begingroup$ Can you elaborate? $\endgroup$
    – LKSR
    Oct 9 '15 at 20:35
  • $\begingroup$ As a sequence in $k$, the limit of $a_{n_k}$ as $k$ goes to infinity is at most the limit supremum, with respect to $k$, of the sequence $a_{n_k}$. Maybe it will help you to write $b_k = a_{n_k}$. There is really nothing mysterious going on here! $\endgroup$
    – Unit
    Oct 9 '15 at 20:43
  • $\begingroup$ Why is the limit at most the limit supremum for any sequence? $\endgroup$
    – LKSR
    Oct 9 '15 at 20:57
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    $\begingroup$ Oh, because for every $n$, we have $a_n \le \sup \{a_m : m \ge n\}$ because $a_n$ is one of the elements of that set. Now take the limit as $n$ goes to infinity: the inequality is still preserved. By definition, $\lim_{n\to\infty} \sup\{a_m: m\ge n\} = \limsup a_n$. $\endgroup$
    – Unit
    Oct 9 '15 at 21:02
  • $\begingroup$ assume for sake of contradiction that limsup $a_n$ $<$ lim $a_n$ = $\alpha$. Then there exists $N$ such that for all $n > N$ we have $|a_n - \alpha| < \alpha - \text{limsup} ~ a_n$. This is a contradiction by the definition of limsup since we have elements larger than it as n tends to infinity. Is this proof correct? $\endgroup$
    – LKSR
    Oct 9 '15 at 21:04

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