Show that $f_n$ converges uniformly to $f=0$

Question

To any integer $n \geq 0$, we associate the function $f_n : [0,1] \to \mathbb R$ defined by $$f_n(x)=x^n(1-x)$$ Show that $f_n$ converges uniformly. I already have a proof involving $\sup\{f_n(x)-f(x)\}$ but I am looking for an alternative proof using something else.

Answer 1

Here is an alternative approach:

If $0 \leq x \leq 1$, we have $x^n \geq x^{n+1}$ and thus $f_n(x) \ge f_{n+1}(x)$ for all $x \in [0,1]$. Clearly we also have $\lim_n f_n(x) = 0$ for all $x \in [0,1]$, so $(f_n)_n$ is a non-increasing sequence that converges pointwise to a continuous function. By Dini's theorem, the convergence is also uniform.