Solution
= Solution
Fix $t_0\in[a,b]$ and define
$$
s(t)=\int_{t_0}^t\lVert\gamma'(u)\rVert,du.
$$
Because $\gamma$ is a <regular curve>, $s'(t)=\lVert\gamma'(t)\rVert>0$. The <inverse function theorem> gives a smooth inverse $t=t(s)$ onto the corresponding interval. For $\widetilde\gamma(s)=\gamma(t(s))$,
$$
\left\lVert\frac{d\widetilde\gamma}{ds}\right\rVert
=\lVert\gamma'(t(s))\rVert\frac{dt}{ds}=1.
$$
Thus the <arc-length parametrization> exists.