Proposition: There is no metric $d: S^2 \times S^2 \to \mathbb{R}$ compatible with the usual topology such that $S^2 - \ast$ is isometric to the Euclidean plane.
My proof: [a gap] it is sufficient to prove that there is no metric $d$ on $\overline{\mathbb{C}}$ such that its restriction on $\mathbb{C}$ is the usual metric on $\mathbb{C}$. Consider a convergent sequence $n \mapsto \frac{1}{n}$. An autohomeomorphism $z \mapsto \frac{1}{z}$ maps it to the sequence $n \mapsto n$, which is not Cauchy in the usual metric on $\mathbb{C}$, and is thus divergent. This is a contradiction.
Is this proof correct? It feels fuzzy in a few places, especially in the first sentence :)