Uniform convergence in real valued injective functions sequence

Question

If $f_{n}$ is a sequence of real valued injective functions that converges uniformily to $f$, then is it necessary that $f$ be injective? Thanks in advance!

Answer 1

Easy counter-example :

Pick $f_n(x):= \frac{x}{n+1}$ for $x\in[0,1],n\in\mathbb{N}$

$\frac{1}{n+1}\ge\lvert{f_n-0}\rvert$ which means that $\frac{1}{n+1}\ge {sup_{x\in[0,1]}\lvert{f_n-0}\rvert}$, which means that $f_n$ converges uniformly towards the trivial function, yet $f_n$ is clearly injective while the zero function clearly isn't