> [!NOTE] Lemma (Closed Euclidean Ball is Closed) > The [[Closed Euclidean Ball|closed Euclidean ball]] is [[Closed Sets|closed]]. ###### Proof Let $y\in \overline{\mathbb{B}_{r}(a)}^C = \{ x\in \mathbb{R}^n : \lVert x-a \rVert > r \}$. Then for all $x\in \mathbb{R}^n$ such that $\lVert x-y \rVert<\varepsilon:=r+\lVert y-a \rVert$, applying the [[Reverse triangle inequality|reverse triangle inequality]] yields $\lVert x-a \rVert \geqslant \big| \lVert y-a \rVert - \lVert y -x \rVert \big| > r \implies x\in \overline{\mathbb{B}_{r}(a)}^C$and so $\overline{\mathbb{B}_{r}(a)}^C$ is open. Since [[Characterization of Open and Closed Subsets of Euclidean Space by Set Complement|complement of open set is closed]], $\overline{\mathbb{B}_{r}(a)}$ is closed. ###### Proof Let $x_{j}$ be sequence in $\overline{\mathbb{B}_{r}(a)}$ such that $\lim_{ j \to \infty } x_{j}=x\in \mathbb{R}$. I