Nested Sphere Theorem

Theorem
Suppose $(M, d)$ is a complete metric space, and suppose $\{S_n[x_n,\rho_n]\}$ is a sequence of closed spheres in $M$ such that $S_1\supset S_2\supset...\supset S_n\supset...$ and $\rho_n\rightarrow 0$ as $n\rightarrow\infty$. Then there exists $x\in M$ such that

\[\bigcap_{n=1}^\infty S_n =x\]

Proof
Let $S_n=S[x_n,\rho_n]$ be the closed sphere of radius $\rho_n$ about the point $x_n$ (i.e. $S_n=\{x\in M:d(x_n,x)\leq \rho_n\}$). Then the sequence $\{x_n\}$ forms a Cauchy sequence: $d(x_n,x_{n+p})<\rho_n$ for any $p\geq 0$ since $S_{n+p}\subset S_n$, however, $\rho_n\rightarrow 0$ as $n\rightarrow\infty$ and therefore $d(x_n,x_{n+p})\rightarrow 0$ as $n\rightarrow\infty$ for any $p\geq 0$. Since the space $M$ is complete, there exists $x\in X$ such that $x_n\rightarrow x$ as $n\rightarrow\infty$. Then since the subsequence $\{x_k\}_{k=n}^\infty$ is contained entirely in $S_n$, converges to $x$ (i.e. is a limit point of $S_n$), and $S_n$ is closed, we have $x\in S_n$ for all $n$. Hence,

\[\bigcap_{n=1}^\infty S_n=x\]