## Math 4A03: Proof of the last lemma on connectedness

Let $(M,d)$ be a metric space; we consider $M^2$ with the corresponding product metric. The goal of this post is to give a proof of the last lemma on these slides: Lemma Let $A,B \subseteq M$ be connected. Then $A \times B$ is connected. The proof uses the exercises preceding this lemma given on the…