# limit superior inequality

That's because $\sup_{m\ge n}(a_m+b_m)\le(\sup_{m\ge n}a_m)+(\sup_{m\ge n}b_m)$. Indeed, on the right you can, roughly speaking, choose the largest elements of the sequences $\{a_m\}$ and $\{b_m\}$ independently and then add them, whereas on the left you have to add elements with the same index and then choose the largest.