Closed Ball is Connected

Theorem
Let $V$ be a normed vector space with norm $\norm {\,\cdot\,}$ over $\R$ or $\C$.

A closed ball in the metric induced by $\norm {\,\cdot\,}$ is connected.

Proof
Let $B^-$ be a closed ball in $V$.

From Closed Ball is Path-Connected:
 * $B^-$ is path-connected in the metric induced by $\norm {\,\cdot\,}$

From Path-Connected Space is Connected:
 * $B^-$ is connected in the metric induced by $\norm {\,\cdot\,}$