Let $A=\{(x,y)\in \mathbb R^2:\max\{|y|,|x|\}\leq 1\}$ and $B=\{(0,y)\in \mathbb R^2:y\in \mathbb R\}$. Show that $A+B$ is a closed subset of $\mathbb R^2$
My try:
let $z_n=x_n+y_n$ be a sequence in $A+B$ converging to $z\in \mathbb R^2$ .To show $z\in A+B$. Now $x_n\in A$
$\implies x_n=(a_n,b_n) $
where $|a_n|,|b_n|\leq 1$and $y_n=(0,c_n)$. Can you help me to proceed further please