If $e^{2 \pi i f_n(x)}$ converges to $e^{2 \pi i f(x)}$ then $f_n$ converges to $f$ uniformly

Question

Let $f_n$ be a sequence of smooth functions on $I=[0,1]$ such that $f_n(0)=0$. Now it is given that for given $\varepsilon>0,\ \exists n_0\in\mathbb{N}$ such that$ \|e^{2 \pi i f_n(x)}-e^{2 \pi i f(x)}\|<\varepsilon$ for every $n\ge n_0$, where $f:\mathbb{R}\to \mathbb{R}$ and $ f(0)=0.$ From this how can I conclude that $f_n\to f$.

Answer 1

hint

$$\|e^{ia}-e^{ib}\|^2=2(1-\cos(a-b))$$

$$=4\sin^2(\frac{a-b}{2})$$