is the image of path metrizable? Let $X$ be Hausdorff and $\alpha$ be a path in $X$.

How do I prove that $\alpha(I)$ is metrizable?

I know that $\alpha(I)$ is compact, connected, locally connected. I tried to prove this by applyin Urysohn's lemma.

Since $I$ is compact metriazble space, there exists a countable dense subset $D$ of $I$.

Fix $t_0\in I$ and $N$ be an open neighborhood of $\alpha(t_0)$. Since $D$ is dense, $\alpha^{-1}(N)\cap D \neq \emptyset$. Thus, $N\cap f(D) \neq \emptyset$. Hence, $f(D)$ is dense in $\alpha(I)$. Thus, $\alpha(I)$ is separable. However, $\alpha(I)$ need not be second countable in this case.

HOw do I prove it?
For $x,y\in \alpha(I)$ let $d(x,y)=\inf\{\,|s-t|:\alpha(s)=x,\alpha(t)=y\,\}$.
For $x,y\in \alpha(I)$ let $d(x,y)=\inf\{\,|s-t|:\alpha(s)=x,\alpha(t)=y\,\}$.