1. ## Compact Sets

Prove that the union of any two compact subsets of the reals is compact.

2. Originally Posted by GoldendoodleMom
Prove that the union of any two compact subsets of the reals is compact.
If $\displaystyle A$ and $\displaystyle B$ are compact sets then there is $\displaystyle R>0$ so that $\displaystyle A,B$ are contained in $\displaystyle \{ x \in \mathbb{R} : |x| < R \}$. But then $\displaystyle A\cup B$ is contained in $\displaystyle \{ x \in \mathbb{R} : |x| < R\}$. Therefore, $\displaystyle A\cup B$ is closed. Also the union of two closed sets is closed therefore $\displaystyle A\cup B$ is closed. Thus, $\displaystyle A\cup B$ is compact.

3. Hello,

Another method is to use the Heine-Borel theorem (note that $\displaystyle \mathbb{R}$ is a separated space) :

If A is a compact, then there exists a finite family of subsets $\displaystyle (A_i)_{i \in I}$, where I is a finite set, such that $\displaystyle A \subseteq \bigcup_{i \in I} A_i$
Similarly, $\displaystyle B \subseteq \bigcup_{j \in J} A_j$, where J is a finite set.

Then $\displaystyle A \cup B \subseteq \bigcup_{k \in I \cup J} A_k$

and $\displaystyle I \cup J$ is finite.

Thus $\displaystyle A \cup B$ is a compact set.