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Let $(f_{n})$ be a sequence of functions $[-1,1] \to \mathbb{R}$ defined by:

$f_{0}(t)=0$ and $f_{n+1}(t)=f_{n}(t)+\frac{1}{2}(t^{2}-f_{n}^{2}(t))$.

Show that $(f_{n})$ converges uniformly. Suggestion: $0\leq f_{n}(t)\leq f_{n+1}\leq|t|$.

I know Dini's theorem that said if $f_{n}$ is defined on compact set, monotonous and converges for a continuous function, then $(f_{n})$ converges uniformly.

From the suggestion we have that $f_{n}$ converges pointwise.

Now I don't know what to do.

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Since

$$0\leq f_{n}(t)\leq f_{n+1}(t)\leq|t|$$

we have that $(f_n(t))$ is non-decreasing and bounded. So there exists $\lim_{n\to \infty} f_n(t).$ Denote $$l=\lim_{n\to \infty} f_n(t)=\lim_{n\to \infty} f_{n+1}(t).$$ Thus taking limits in $$f_{n+1}(t)=f_{n}(t)+\frac{1}{2}(t^{2}-f_{n}^{2}(t))$$ we get

$$l=l+\dfrac 12 (t^2-l^2).$$ So it is $l^2=t^2.$ Since $l\ge 0$ it must be $l=|t|.$

We have shown that the sequence $(f_n)$ converges pointwise to the continuous function $f(t)=|t|.$ Using Dini's theorem we get that the convergence is uniform on $[-1,1].$

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