Closed Unit Ball is Convex Set

Theorem
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Let $\map {B_1^-} 0$ be the closed unit ball in $X$.

Then $\map {B_1^-} 0$ is convex.

Proof
Let $x, y \in \map {B_1^-} 0$.

Let $\alpha \in \closedint 0 1$ be arbitrary.

Then:

Therefore, $\paren {1 - \alpha}x + \alpha y \in \map {B_1^-} 0$.

By definition, $\map {B_1^-} 0$ is convex.