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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?

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