Sum of two compact sets is compact

Question

I didn't find an already existing proof for the following question:

We have two compact sets $A, B \subset \Bbb R^n$. I want to show that the addition of these is also compact.

So, I want to show $A+B:=\{a+b: a\in A, b\in B \}$ is compact.

Please help me.

Answer 1

Proof using sequential compactness: Suppose $a_k+b_k \to y$ where $a_k \in A$ and $b_k$ in $B$ for all $k$. There exists a subsequence $a_{k_i}$ converging to some point $a \in A$. Now look at $b_{k_i}$. This has a subsequence $b_{k_{i_l}}$ converging to some $b$ in $B$. Can you now show that $y=a+b$?

Answer 2

• The set $A\times B\subset{\mathbb R}^{2n}$ is compact.
• The map $\quad {\rm add}: \ A\times B\to {\mathbb R}^n,\quad (a,b)\mapsto a+b\quad $ is continuous.