ILL 1974 Problem 33

Let a be a real number such that $0 < a < 1$ , and let $n$ be a positive integer. Define the sequence $a_0, a_1, a_2, \ldots, a_n$ an recursively by $$a_0 = a, a_{k+1} = a_k +\frac 1n a_k^2 \text{ for } k = 0, 1, \ldots, n - 1.$$ Prove that there exists a real number $A$ , depending on $a$ but independent of $n$ , such that $$0 < n(A - a_n) < A^3.$$