Show that the open balls and closed balls in the standard metric on $\mathbb{R}^n$ are connected in $\mathbb{R}^n$
Show that the open balls and closed balls in the standard metric on $\mathbb{R}^n$ are connected in $\mathbb{R}^n$
note that $S \subseteq \mathbb{R}^n$ is path-wise connected, implies that it is connected. thus, you have the desired result, if you can show that open balls and closed balls are path-wise connected.