To show sum of two sets is closed in R^2 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

$A$ is compact and $B$ is closed. Now it is a general fact that (in Euclidean spaces) in this case $A+B$ is closed (try to prove this before reading further).
In fact, take any sequence $p_n=a_n+b_n\to p\in\mathbb{R}^2$, where $a_n\in A$ and $b_n\in B$. We have to prove that $p\in A+B$. 
As $A$ is compact, upon possibly passing to a subsequence, we have $a_n\to a\in A$. 
So $b_n=p_n-a_n\to p-a$ and since $B$ is closed we have $p-a\in B$. 
Call $b:=p-a$; we just proved that $p=a+b$ with $a\in A$ and $b\in B$, so we are done.
