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## Integral recurrence

Suppose $y_1(x)$ is an arbitrary, continuous, positive function on $[0,A]$, where $A$ is an arbitrary positive number. Consider the following recurrence:
$\displaystyle y_{n+1}(x)=2 \int_0^x \sqrt{y_n(t)}dt,\ n \geq 1$.
Prove that the sequence $(y_n)$ converges uniformly to $y=x^2$ on $[0,A]$.
Miklos Schweitzer 1964

Hint: What happens when $y_1$ is constant?