A Closed Euclidean Ball is Convex and Compact

Let $n$ be a natural number with $1\le n$, let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure, let $x$ be a point of Euclidean space $\mathbb{R}^n$, regarded as a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, and let $r\in\mathbb{R}$ satisfy $0\le r$. Let $d_E$ be the Euclidean distance, a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$, let $\mathcal{T}_{d_E}$ be the collection of subsets of $\mathbb{R}^n$ that are open in $(\mathbb{R}^n,d_E)$, a topology by Metric Open Sets Form a Topology, and let $\bar{B}_{d_E}(x,r)$ be the closed ball in $(\mathbb{R}^n,d_E)$.

Then the following hold.

1. (Convexity) $\bar{B}_{d_E}(x,r)$ is a convex subset of $\mathbb{R}^n$.

2. (Compactness) $\bar{B}_{d_E}(x,r)$ is compact in $(\mathbb{R}^n,\mathcal{T}_{d_E})$.