Does $f_n$ uniformly converge to $f$? $f_n :[0, 1]\rightarrow\mathbb{R} \qquad x \mapsto x^n - x^{n+1}$
The sequence converges pointwise to the zero function. It converges uniformly if
$$\sup_{x \in [0, 1]} \; \big| \, f_{n}(x) - f(x) \, \big|$$
tends to zero. But I am not sure if it does or how to prove.

Notice that $f_{n+1} = x f_n$ as $x\in[0,1]$ you have that $f_n$ is monotone. Use Dini's theorem Link to complete the proof.
