Problem: Prove that the union of any collection of open sets is open. Is the following proof acceptable?

Proof: Let U' denote the collection of open sets and let U' =

, where n = 1,...,k. Since U' is a collection of open sets, there exists an interior point

that belongs to U'. Since

belongs to U', it is implied that

belongs to at least one

. So for

that belongs to

, there exists a neighborhood in

; For

that belongs to

, there exists a neighborhood in

,...If we take the union of all

, there exists a neighborhood in U' for all

. Therefore, U' is open.