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Problem 56: (Unions And Intersections Of Finitely Many Opens Sets Are Open)
Use induction to prove for each $n\in \mathbb{N}$ if $U_1$, $U_2$, ..., $U_n$ are open sets, then $\ds \bigcup_{i=1}^n U_i$ is an open set.
A similar proof will show that for each $n\in \mathbb{N}$ if $U_1$, $U_2$, ..., $U_n$ are open sets, then $\ds \bigcap_{i=1}^n U_i$ is an open set.
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