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Math Help - Compact Sets

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    Compact Sets

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

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

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

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

    and I \cup J is finite.

    Thus A \cup B is a compact set.
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