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I have a question of finding lim sup and lim inf of $a_n=\frac{1}{n} + (-1)^n$ and prove $\liminf a_n \leq \limsup a_n.$ So the work below is what I did for the first part.

$a_{odd\ n} = \frac{1}{n}-1$ and $a_{even\ n} = \frac{1}{n}+1.$ So $\limsup a_n = 1$ and $\liminf a_n = -1.$ How do I prove the second part?? I tried to use the definition but I am confused with the definition.

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    $\begingroup$ It must be the other way around, $a_{odd}=\frac{1}{n}-1$ and $a_{even}=\frac{1}{n}+1$ $\endgroup$
    – the8thone
    Dec 19, 2013 at 2:26
  • $\begingroup$ @Roozbeh-unity tahnks.. fixed!! $\endgroup$
    – eChung00
    Dec 19, 2013 at 2:41
  • $\begingroup$ See also: math.stackexchange.com/questions/353642/… (and other questions shown there among linked questions). $\endgroup$ Jan 29, 2015 at 20:12

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Here is how to prove $\liminf \{ x_n \} \leq \limsup \{ x_n \}$ for a general real sequence $\{ x_n \}$. Set $L = \liminf \{ x_n \}$ and $S = \limsup \{ x_n \}$, and suppose for the sake of contradiction that $L > S$. Say $L = S + h$ for some $h > 0$. Then there are infinitely many $n$ for which $x_n \in (L - h, L + h)$. Therefore, there are infinitely many $n$ for which $x_n > S$, contradicting the definition of $S$. Hence, $L \leq S$.

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I suppose if you can see that $\liminf a_n = -1$ and $\limsup a_n = +1$, then trivially $\liminf a_n \leq \limsup a_n$.

Indeed for "any" sequence that is the case, the definition forces that, so it is among properties of limsup and liminf.

From Wikipedia http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior

Whenever $\liminf x_n$ and $\limsup x_n $both exist, we have $\liminf_{n \to \infty}x_n\leq \limsup_{n \to \infty}x_n$.

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  • $\begingroup$ I thought I have to prove it somehow..thanks! $\endgroup$
    – eChung00
    Dec 19, 2013 at 2:43

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