Prove $x(1-\frac{1}{2}x)^n\rightarrow 0$ uniformly

Question

Let $x\in[0,1]$, and define $f=x(1-\frac{1}{2}x)^n$, where $n\in\mathbb{N}$. Certainly $x(1-\frac{1}{2}x)^n\rightarrow 0$ as $n\rightarrow\infty$. How to prove that this convergence is uniform by it is bounded by $\frac{2}{n+1}$?

Answer 1

Let $f_n=x(1-\frac12 x)^n$

The following are true:

• $[0,1]$ is compact,
• $f_n(x) \ge f_{n+1}(x)$ (check this)
• Each $f_n$ is continuous, and $f_n(x) \to 0$ for all $x$ (note that $0$ is continuous)

Therefore you may apply Dini's Theorem to conclude the convergence is uniform.