Catalog of theorems from Real Analysis alongside proofs for each.

Proposition 8. Unions and Intersections of Closed Sets
§4.3
If A,B⊂Rn are closed, then A∪B is closed.If {Ai:i∈I} is a family of closed subsets of Rn,then ⋂i∈IAi is closed.\text{If } A, B \subset \mathbb{R}^n \text{ are closed, then } A \cup B \text{ is closed.} \\ \text{If } \{A_i : i \in I\} \text{ is a family of closed subsets of } \mathbb{R}^n, \\ \text{then } \bigcap_{i \in I} A_i \text{ is closed.}

Lemma 9. Closure is the Smallest Closed Set
§4.3
Let A⊆Rn. Then A‾ is the smallest closed set containing A.In particular, A‾‾=A‾.\text{Let } A \subseteq \mathbb{R}^n. \text{ Then } \overline{A} \text{ is the smallest closed set containing } A. \\ \text{In particular, } \overline{\overline{A}} = \overline{A}.

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