general topology – Convergence of $f o g_n$ towards $fo g$

Let $fin C^1((0,1)timesmathbb{R},mathbb{R})$, and $g_n in C^1((0,1),mathbb{R})$ that converges uniformly towards $g in C^1((0,1),mathbb{R})$ : i.e
$$||g_n-g ||_{infty}rightarrow0. ,, (nrightarrow +infty).$$
Can we deduce that :for all $tin (0,1)$ : $|f(g_n(t),t)-f(g(t),t)|rightarrow0. ,, (nrightarrow +infty)$ ?

My idea:

Since $fin C^1$, then $f$ is locally lipschitz : there is $c>0$ such that :
$$|f(g_n(t),t)-f(g(t),t)|leq c ||g_n-g||_{infty}.$$