The elements of a closed set and compact set form a closed set.

Question

Let $A$ be a compact subset and $B$ be a closed subset in $\mathbb{R}^n$.

Prove that the subset $\alpha A+B$ (which is defined as $\{ \alpha a+b: a \in A, b \in B, \alpha \in [-3,2] \}$) is a closed subset of $\mathbb{R}^n$.

My Attempt: First $\alpha$ is a closed set and $A$ is closed by the Heine-Borel theorem.

Let $x$ be a limit point of $\alpha A+B$. Then there exists a sequence $\{p_i: i\in \mathbb{N} \}$ in $\alpha A+B$ which converges to $x$. Now suppose $x \not \in \alpha A+B$. Let $\epsilon = \Vert x- (\alpha a+b) \Vert $. Since $\lim_{i\rightarrow \infty}p_i=x$, there exists $N$ such $i>N$ implies $\vert x - p_i\Vert < \frac{\epsilon}{2}$. Since $\Vert x- p_i \Vert \geq \epsilon$ we have a contradiction. Then $x$ must belong to $\alpha A+B$ . Hence as $\alpha A+B$ contains its limit points it is closed.

Any comments on my anwser would be appreciated.

Answer 1

Let $M = \{ \alpha a + b \mid \alpha\in [-3,2], a\in A, b\in B \}$, $x_n\in M$ and $x_n\to x$.

Hints:

1. There exist $\alpha_n\in [-3,2]$, $a_n\in A$, and $b_n\in B$ with $x_n = \alpha_n a_n + b_n$.
2. As $[-3,2]$ and $A$ are compact, there are convergent subsequences, say $\alpha_{n_k}\to \alpha^* \in [-3,2]$ and $a_{n_k} \to a^* \in A$.
3. Show that the subsequence $b_{n_k}$ converges and has limit $b^*\in B$.
4. Thus, $x = \alpha^* a^* + b^* \in M$.