I have been trying to come up with a proof, but I don't think I am making leaps in logic and assuming things that I'm possibly trying to prove, if anyone could help correct/(or completely discard) what I have done so far and help me towards a rigerous proof I would greatly appreciate it. This is what I have come up with so far.
Let $U_1,U_2,...,U_n$ be subsets of $\mathbb{R^{n}}$ and closed, define
$U:=\bigcap _{i=1}^nU_i$
And the boundary of $U$ as $B:=CL(U)\cap CL(\mathbb{R^{n}}-U)$
WLG take $U_1,U_2,...,U_m$ where $m\leq n$ such that for some $W_1,W_2,...,W_m$ where $W_i\subset U_i$
$B:=\bigcup _{i=1}^mW_i$
Take $x\in B$ $\Rightarrow x\in \bigcup _{i=1}^mW_i$ $\Rightarrow x\in U_i$ for some $i=1,2,...,m$ $\Rightarrow x\in \bigcap _{i=1}^nU_i \Rightarrow x\in U $
Therefore $B\subset U$ therefore $U$ is closed
Pretty sure this is wrong, but I thought it would be good to show you my train of thought