If $A \subset \mathbb{R}^n$ is star domain show that it is connected set.

Question

If $A \subset \mathbb{R}^n$ is star domain show that it is connected set.

So my idea is to show that $A$ is path-connected. Let $x,y \in A$. There exists $x_0 \in A$ such that $[x,x_0] \subset A$ and $[x_0,y] \subset A$. We can create continuous function $\alpha:[0,1] \rightarrow A$ such that $\alpha(0)=x, \alpha(0.5)=x_0$,$\alpha(1)=y$ and $\alpha([0,1])=\{ [x,x_0] \cup [x_0,y] \}$. Therefore, $A$ is path-connected so it connected. Is this valid proof?