[Math] If $A$ is closed and $B$ is compact in $\mathbb R^n$ then $A+B=\{a+b : a \in A \text{ and } b \in B\}$ is closed.

Question

If $A$ is closed and $B$ is compact in $\mathbb R^n$ then $A+B=\{a+b : a \in A \text{ and } b \in B\}$ is closed. (In other words, the vector/Minkowski sum of a closed set and a compact set is closed.)

What I've tried so far:
Let $ c_n $ be a sequence in $A+B$; $c_n =a_n+b_n$ where $a_n \in A$ and $b_n \in B$. Since $B$ is compact, there exists a subsequence $(b_{n_k})$ which converges $b$ which is in $B$.

Now I'm stuck in how to show that the subsequence $(a_{n_k})$ converges to some number in $A$ so that $(c_{n_k})$ converges to the sum of two limits in $A+B$.

Answer 1

As commenters indicated, a typical proof proceeds as follows:

1. Assume that $(c_n)$ is a convergent sequence in $A+B$.
2. Write $c_n=a_n+b_n$.
3. Pick a convergent subsequence $b_{n_k}\to b\in B$.
4. Note that $a_{n_k}=c_{n_k}-b_{n_k}$ is also convergent. Let $a$ be its limit.
5. Conclude that $\lim c_n=\lim {c_{n_k}} = a+b \in A+B$.