I want to show that the function:
$A: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n, (x, y) \mapsto x + y$
is continuous.
Also, why is it that if $K, L$ are compact subspaces of $\mathbb{R}^n$, $K + L = \{x + y | x \in K, y \in L\}$ is also compact?
Thanks in advance. Intuitively it sounds right. For proving it, maybe one could use the tube lemma here?
Your phrasing of the problem is wrong, assuming that you wanted to say $A: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ and it is defined as the addition of the vector space $\mathbb{R}^n$, then we will show that for any sequence $a_k = (x_k,y_k) \in \mathbb{R}^n\times \mathbb{R}^n$ converging to $a=(x,y)$ we will show that $A(a_k) $ converges to $A(a)$ which will establish the continuity. Now $a_k $ converges to $(x,y)$ means $x_k$ and $y_k$ converges to $x$ and $y$ respectively. So $A(a_k)=x_k+y_k$ converges to $x+y$ since sum of two convergent sequences is convergent and sandwich theorem will tell us that the limit is the desired one.
For the second part note that $K \times L$ is compact in $\mathbb{R}^n\times \mathbb{R}^n$ by Tychonoff's theorem. And $K+L= A(K\times L)$. Now since continuous image of a compact set is compact, the required set is compact.