To bkarpuz, about your 1st post: I feel like you only prove $\displaystyle c_n\leq \max(a_n,b_n)$ in a complicated way, or maybe I missed something? Then you can deduce $\displaystyle \limsup_n c_n\leq \limsup_n \max(a_n,b_n)$.

It is a true fact that $\displaystyle \limsup_n \max(a_n,b_n)=\max(\limsup_n a_n,\limsup_n b_n)$, but not a trivial one. Actually this is a particular case of the initial problem, and the answer Opalg gave to it gives also a proof of this fact (it proves the inequality "$\displaystyle \leq$", and "$\displaystyle \geq$" is almost trivial).

Here's a variant of Opalg's proof, maybe more complicated, but maybe more intuitive: there exists a subsequence $\displaystyle (c_{\phi(n)})_{n\geq 0}$ which converges toward $\displaystyle \limsup_n c_n$. For any $\displaystyle n$, either $\displaystyle c_{\phi(n)}\leq a_{\phi(n)}$ or $\displaystyle c_{\phi(n)}\leq b_{\phi(n)}$, so that one case happens for infinitely many $\displaystyle n$. For instance, there are infinitely many $\displaystyle n$ such that $\displaystyle c_{\phi(n)}\leq a_{\phi(n)}$. Then we can pick a subsequence $\displaystyle (a_{\phi(\psi(n))})_{n\geq 0}$ of $\displaystyle (a_{\phi(n)})_{n\geq 0}$ which converges toward a limit $\displaystyle \ell\in\mathbb{R}\cup\{\pm\infty\}$ (for instance, you can take $\displaystyle \ell=\limsup_n a_{\phi(n)}$). From $\displaystyle c_{\phi(\psi(n))}\leq a_{\phi(\psi(n))}$, we deduce (taking the limsups)

$\displaystyle \limsup_n c_n\leq \ell\leq \limsup_n a_n\leq \max(\limsup_n a_n,\limsup_n b_n)$.

(we have $\displaystyle \ell\leq \limsup_n a_n$ because the limsup is the maximum of the limits of subsequences, a fact I already used before. )

Originally Posted by

**Showcase_22**

I'm not really sure what to put for this part of the question. It seems like nothing conclusive can be written.

Does anyone have any ideas?

Well, you could say

$\displaystyle \min(\liminf_n a_n,\liminf_n b_n)\leq \liminf_n c_n\leq\limsup_n c_n \leq \max(\limsup_n a_n, \limsup_n b_n)$,

but you're right, you can't give any lower bound with limsups.