Prove that $\limsup s_n + \liminf t_n \leq \limsup (s_n + t_n)$. $s_n$ and $t_n$ are bounded.
I don't know how to start this, can anyone help and give a formal proof of this?
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Martin Sleziak
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mathcoming
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What is your definition of $\liminf , \limsup$? There are many equivalent definitions. – Crostul Mar 25 '16 at 13:11
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The best help, one can give you for such an homework question, would be a hint: Look up the definitions of lim inf and lim sup and put them into the situation. Then the problem will basically solve itself. – CandyOwl Mar 25 '16 at 13:12
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1The first thing that you should do is correct the statement of the theorem, because what you have here is false. Suppose that $s_n=(-1)^n$ and $t_n=(-1)^{n+1}$. Then $\limsup_ns_n=1=\limsup_nt_n$, but $s_n+t_n=0$ for all $n$, so $\limsup_n(s_n+t_n)=0$, and $1+1\not\le 0$. – Brian M. Scott Mar 25 '16 at 14:02