when does $\limsup(a_n + b_n) < \limsup(a_n) + \limsup(b_n)$ holds? I proved the inequality for $\limsup(a_n + b_n) \le \limsup(a_n) + \limsup(b_n)$
but when is the strict inequality true given as  $\limsup(a_n + b_n) < \limsup(a_n)+ \limsup(b_n)$
for me, i think it may be true that if $\limsup(a_n) + \limsup(b_n)$ both diverge individually, but the $a_n +b_n$ sequences converge, then this strict inequality holds.  but is this the only case? 
Thank you very much

$\limsup a_n$ and $\limsup b_n$ don't have to be infinite - it is sufficient that $a_n$ is small when $b_n$ is big and conversely. 
One simple example is
$a_n = (-1)^n \\ 
b_n = -(-1)^n$
with 
$\limsup a_n = \limsup b_n = 1$ and $\limsup( a_n+b_n ) = 0.$
