# 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 :
$$|f(g_n(t),t)-f(g(t),t)|leq c ||g_n-g||_{infty}.$$