Let $A$, $B$ be compact subsets of $\Bbb R$ and define $C = A + B =\{\, a+b\mid a \in A , b \in B\,\}$.

Question

Let $A$, $B$ be compact subsets of $\Bbb{R}$ and define $C = A + B =\{\, a+b\mid a \in A , b \in B\,\}$. Prove that $C$ is compact. Use induction to show if $A_1, A_2,\ldots A_n$ are compact subsets of $\Bbb{R}$, then $A_1 +A_2 + \ldots+ A_n$ is compact.

Proof.

Since $A$ and $B$ are compact subsets of $\Bbb{R}$, we know they are both bounded and closed.

Thus, there exists $\alpha \in A$ and $\beta \in B$ such that $\alpha \ge a_i$ for any $a_i$ in $A$ and likewise $\beta \ge b_i$ for any $b_i$ in $B$. Let $\gamma = \alpha + \beta$, then $C$ is bounded by $\gamma$. Since $C$ is the union of a finite number of closed sets, then $C$ is also closed, thus, since $C$ is closed and bounded, it is compact.

Is this correct for the first part?

Answer 1

Note that $+\colon \Bbb R^2\to\Bbb R$ is continuous, hence the image $C$ of the compact set $A\times B\subset \Bbb R^2$ is compact.

Answer 2

For first part, you are right on $A$ and $B$ are compact, they are both bounded and closed.

Since $A$ and $B$ are bounded, so is $A+B$. This is because for any $a\in A$, there is $M_A$ that $|a|<M_A$. For any $b\in B$, there is $M_B$ that $|b|<M_B$. So $|a+b|<M_A+M_B$.

Let $c_n\to c,\:c_n\in A+B$. Then $c_n=a_n+b_n,\:a_n\in A,b_n\in B$. Since $A$ and $B$ are closed, the limit point of $a_n$ (if it exists) is in $A$ and limit point of $b_n$ (if it exists) is in $B$. So the limit point of $a_n+b_n$ is in $A+B$. This proves that $A+B$ is closed.

Since $A+B$ is closed and bounded, it is compact.