Let $\displaystyle U\subseteq R^n$ be an open set if $\displaystyle \forall x\in U $ there exists an r > 0 such that B(x,r) $\displaystyle \subseteq U$. B(x.r) meaning a ball centered on x with radius r.

1) Let a $\displaystyle \in R^n$. Prove that for all $\displaystyle \epsilon > 0 $ B(a,$\displaystyle \epsilon$) is open

2) Show that if $\displaystyle U_1, .... ,U_n$ are open in $\displaystyle R^n$ then $\displaystyle U_1\cap ..... \cap U_n$ also open