Let an, bn be two bounded sequences. Show that
limn-->∞ sup (an+bn) ≤ limn-->∞ sup an + limn-->∞ sup bn
That's because $\displaystyle \sup_{m\ge n}(a_m+b_m)\le(\sup_{m\ge n}a_m)+(\sup_{m\ge n}b_m)$. Indeed, on the right you can, roughly speaking, choose the largest elements of the sequences $\displaystyle \{a_m\}$ and $\displaystyle \{b_m\}$ independently and then add them, whereas on the left you have to add elements with the same index and then choose the largest.