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I'm going through lecture notes and I've come to a point that says, for real sequences $a_n, b_n$, that $\limsup \max (a_n, b_n) \leq \max(\limsup a_n, \limsup b_n)$. Can we do better and get a sharper "bound" and actually have an equality instead?

I simply assumed that we could interchange $\limsup$ and $\max$ without a problem, and the $\leq$ direction is obvious, but I don't know how to prove the $\geq$ direction.

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By definition $\limsup$ is increasing and $a_n\le \max(a_n, b_n)$ therefore: $$\limsup \max(a_n, b_n) \ge\limsup a_n $$ Similiraly :

$$\limsup \max(a_n, b_n) \ge\limsup b_n$$

Thus

$$\limsup \max(a_n, b_n) \ge\max(\limsup b_n, \limsup a_n)$$

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  • $\begingroup$ Why the downvote ? Could you please elaborate ? $\endgroup$ – Astyx Apr 22 '17 at 11:48

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