How to prove that $\;\limsup\limits_{n\to\infty}X_n \leq \sup\limits_{n\in\mathbb{N}}\{X_n\}\;$?
I need to prove this and I don't know how to go about doing this.
Thank you for any help you can provide.
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limcp? Do you mean $\limsup$? – Plutoro Apr 07 '16 at 01:31
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It is not clear what you are asking. – copper.hat Apr 07 '16 at 01:32
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Yes limsup{Xn} <= sup{Xn} sorry I will edit. – Matt Apr 07 '16 at 01:33
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Could you also edit your question to be more clear, and show what you know? As it stands, your question may be closed for being either unclear or lacking detail. – Plutoro Apr 07 '16 at 01:36
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By definition $$\limsup\limits_{n\to\infty}X_n =\inf_k \sup_{n\geq k}X_n.$$ But of course $$\sup_{n\geq k}X_n \leq \sup_n X_n$$ for all $k$. Hence $$\inf_k \sup_{n\geq k}X_n \leq \sup_n X_n.$$
Angelo
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C. Dubussy
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Alternative proof:
$x=\limsup\limits_{n\to\infty}X_n\;$ only if a subsequence of $\;\left\{X_n\right\}_{n\in\mathbb{N}}$ approaches $\;x\;$ and this subsequence is bounded above by $\sup\limits_{n\in\mathbb{N}}X_n$.
Angelo
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Henricus V.
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