Is it true that $\limsup \max (a_n, b_n) = \max(\limsup a_n, \limsup b_n)$?

Question

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.

Answer 1

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

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