Prove that if $S$ and $T$ are subspaces of $V$, then so is $S\cap T$
1) Since S, T are subspaces of V, the unique zero vector is contained in both, and so the zero vector is in the intersection.
2) Let $a,b \in S \cap T$. That means a, b are in both, S and T. Since $S,T$ are subspaces, they are closed under addition and so $a+b$ is in S and in T and so in the intersection.
3) Let c be an element of the reals. Let a be in the intersection of $S,T$. Since, $S,T$ are both subspaces, It follows that $ca$ is in both and therefore in the intersection.