$A,B$ bounded $\Rightarrow$ $A+B$ bounded
Solution 1:
Let $U$ be a neighbourhood of $0$. Choose a balanced neighbourhood $V$ of $0$ such that $V+V\subset U$. Pick $n_1,n_2 > 0$ such that $A\subset \lambda V$ for $\lvert\lambda\rvert \geqslant n_1$ and $B\subset \lambda V$ for $\lvert \lambda\rvert \geqslant n_2$. Let $n = \max\{n_1,n_2\}$. Then for $\lvert\lambda\rvert \geqslant n$ we have ...