These are definitions given from Barret O'neill's Elementary Differential Geometry.
Definition $1$. A curve in $\mathbb{R}^3$ is a differentiable function $\alpha: I\to \mathbb{R}^3$ from an open interval $I$ into $\mathbb{R}^3$.
Definition $2$. Let $\alpha: I\to \mathbb{R}^3$ be a curve. If $h: J\to I$ is a differentiable function on an open interval $J$, then the composite function
$$\beta=\alpha (h):J\to \mathbb{R}^3$$
is a curve called a reparametrization of $\alpha$ by $h$.
My question is, how can I prove that any two unit speed reparametrization, $\alpha(h)$ is essentially a reparametrization by the arc length function $s(t)=\int_{t_i}^t ||\alpha '(u)||du$, with only different base points, say $t_1$ and $t_2$.
To solve this problem, it's enough to show that any unit speed reparametrization $\beta=\alpha(h)$ of $\alpha$ is reparametrized by an arc length function at some base point $t_0\in I$. However, the only fact I have is $||\beta'(w)||=||\alpha '(h(w))|||h'(w)|=1$. I need to show that $h$ is an inverse function of the arc length $s$ of $\alpha$. But I have no idea how to progress from here.
I would greatly appreciate it if anyone could explain this to me, I'm having a hard time grasping this.