Why is a curve parameterized by arc length necessarily a unit speed curve?

Question

I apologize if this is trivial but I have not been able to figure it out. For a curve $\sigma(t)$, I have a definition for arc length:

$$s(t)=\int_{t_0}^t |\sigma'(t)|dt$$

We reparameterize a curve $\sigma$ by observing that $s$ has an inverse, $t(s)$, and the resulting reparameterized curve has underlying assignment $s \mapsto t \mapsto \sigma(t)$.

A curve is a unit speed curve if

$$\forall t\qquad |\sigma'(t)|=1$$

Thanks

Answer 1

If $t=s$ your definition of arc-length becomes $$ s=\int_{s_0}^s|\sigma'(u)|\,du $$ Differentiating w.r.t $s$ we get $$ 1=|\sigma'(s)| $$ which shows when parametrized using $s$, the curve becomes unit speed.