State and prove the Baire Category Theorem. [Choose any version you like.]

An isometry from a metric space $(M, d)$ to another metric space $(N, e)$ is a function $\varphi: M \rightarrow N$ such that $e(\varphi(x), \varphi(y))=d(x, y)$ for all $x, y \in M$. Prove that there exists no isometry from the Euclidean plane $\ell_{2}^{2}$ to the Banach space $c_{0}$ of sequences converging to 0 . [Hint: Assume $\varphi: \ell_{2}^{2} \rightarrow \mathrm{c}_{0}$ is an isometry. For $n \in \mathbb{N}$ and $x \in \ell_{2}^{2}$ let $\varphi_{n}(x)$ denote the $n^{\text {th }}$coordinate of $\varphi(x)$. Consider the sets $F_{n}$ consisting of all pairs $(x, y)$ with $\|x\|_{2}=\|y\|_{2}=1$ and $\|\varphi(x)-\varphi(y)\|_{\infty}=\left|\varphi_{n}(x)-\varphi_{n}(y)\right|$.]

Show that for each $n \in \mathbb{N}$ there is a linear isometry $\ell_{1}^{n} \rightarrow \mathrm{c}_{0}$.