Euclidean Balls are Convex

Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the ordered field of real numbers. Equip Euclidean space $\mathbb{R}^n$ with the Euclidean distance $d_E$, a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$. Let $a\in\mathbb{R}^n$ and let $r\in\mathbb{R}$ satisfy $0<r$.

Then both of the following subsets of $\mathbb{R}^n$ are convex.

1. (Open ball) The open ball $B_{d_E}(a,r)$.

2. (Closed ball) The set

$$\overline{B}(a,r)=\{y\in\mathbb{R}^n:\ d_E(a,y)\le r\}.$$