Show that $\overline{K_r(x)} = \overline{K}_r(x)$.

Show that $\overline{K_r(x)} = \overline{K}_r(x)$.

• $K_r(x)$ denotes the open ball of radius $r$ centered at $x$
• $\overline{K_r(x)}$ is the closure of $K_r(x)$, which is the set of all points in $K_r(x)$ together with its boundary
• $\overline{K}_r(x)$ denotes the closed ball of radius $r$ centered at $x$

I also know that $\overline{K_r(x)}$ includes all points in $K_r(x)$ and all limit points of sequences in $K_r(x)$. Any idea on how to prove this statement in any direction? Thanks.