# Statement(s) > [!NOTE] Statement 1 (Open Ball is Open) > The [[Open Euclidean Ball|open ball]] given by $B_{r}(a)=\{ x\in \mathbb{R}: \Vert x-a \Vert <:r \}$where $r>0$ and $a\in \mathbb{R}^n$ is an [[Open subsets of metric spaces|open set]]. **Proof**: For each $y \in \mathbb{B}_{r}(a)$ we need to find $\rho_y>0$ so that the open ball $\mathbb{B}_{\rho_{y}}\left(y\right) \subset \mathbb{B}_{r}(a)$. To this end, set $\rho_y=r-|y-a|$. Then, since $|y-a|<r$, we have $\rho_y>0$ and, for $x \in \mathbb{B}_{\rho_{y}}\left(y\right)$,$|x-a| \leqslant|x-y|+|y-a|<\rho_y+|y-a|=r$