Let $\Omega$ be a smooth bounded open domain in $\mathbb R^d$. Let $\gamma_n: [0,1]\to \overline{\Omega}$ be a sequence of Lipschitz functions which converges uniformly on $[0,1]$ to a Lipshitz function $\gamma : [0,1]\to \overline{\Omega}$.
Does one have for all $t\in [0,1]$
$$\liminf \vert \gamma_n' (t)\vert \geq \vert \gamma'(t)\vert,$$
where ' stands for $d/dt$. (?)
And if not, does one have a counterexample ?
Thanks for your answers