Let ${f_n}$ be a sequence of polynomial with real coefficients defined by $f_0 = 0$ and for $n= 0,1,2,\ldots$ $$f_{n+1}(x) = f_n(x) + \frac {{x^2} -f_n^2(x)}{2}$$ Find $\lim_\limits{n\rightarrow \infty}f_n$ on $[-1,1]$, where the limit is taken in the supremum norm of $f_n $ over the interval $[-1,1]$.
My answer is $\lim\limits_{n\rightarrow \infty}f_n$ $=0$ as it already given that $f_0 =0$
EDIT: $$f_1 = f_0+ \frac {x^2 -f_0}{2}$$ as $f_1 = \frac {x^2}{2} $ and, similarly, $f_2 =x^2 -\frac{x^2}{8}$
As I am not able to proceed further as I don't know how to approach this question, please help me
Suppose $\lim_\limits{n\to\infty}f_n=l,$ then also $\lim_\limits{n\to\infty}f_{n+1}=l$. So $$f_{n+1}(x) = f_n(x) + \frac {{x^2} -f_n^2(x)}{2}\implies l=l+\frac{x^2-l^2}{2}$$