Nested sequence of non-empty compact subsets - intersection differs from empty set

Suppose $x \in S_1$. As we assumed at the start that $\bigcap_{n=1}^\infty S_n =\emptyset$, $x$ cannot be a member of all $S_n$ so some $n_0$ exists such that $x \notin S_{n_0}$, or equivalently $x \in V_{n_0}$. So the $V_n$ cover $S_1$.

Note that as the $S_n$ are decreasing, so the $V_n$ are increasing, so when we have a finite subcover, the one with the largest index (which exists by finiteness) is a superset of all of them, so we have a subcover consisting of one set, say $V_k$. But this cannot be, as any point of $S_k$ (which exists, as these sets are non-empty) is by definition not covered by $V_k$. This contradiction shows that the original assumption of $\bigcap_{n=1}^\infty S_n = \emptyset$ was false.