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I have to show that the polynomial sequence :

$P_n : [0, 1] \rightarrow \Bbb R, P_0(x) = 0$ and $P_{n+1}(x) = P_n(x) + \frac{1}{2}(x-P_n^2(x))$, for each integer $n \ge 0$

converges uniformly to $\sqrt{x}$.

In order to do that, I have to use Dini's theorem. I've already proved that :

  • For all $x \in [0, 1]$ and for all integer $n \ge 0$ : $0 \le P_n(x) \le P_{n+1}(x) \le \sqrt{x}$

  • $P_n(x)$ and $\sqrt{x}$ are continuous on $[0, 1]$

Now I only need to show that $P_n(x)$ converges pointwise, for all $x \in [0,1]$, to $\sqrt{x}$, but I didn't succeed in doing it with the definition of pointwise convergence.

Any help ? Thanks.

Desura
  • 2,011

1 Answers1

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If $x\in [0,1]$, then $(P_n(x))$ converge (bounded by $\sqrt x$ and increasing). Let $$f(x)=\lim_{n\to\infty }P_n(x).$$ Therefore, $$f(x)=f(x)+\frac{1}{2}(x-f^2(x))\implies f^2(x)=x\underset{P_n\geq 0}{\implies} f(x)=\sqrt x.$$

And now you can use Dini

Surb
  • 55,662