Putnam 1985 A3

Let $d$ be a real number. For each integer $m \geq 0$, define a sequence $\{a_m(j)\}$, $j=0,1,2,\dots$ by the condition

$$\begin{aligned}a_m(0) &= d/2^m, \\ a_m(j+1) &= (a_m(j))^2 + 2a_m(j), j \geq 0.\end{aligned}$$

Evaluate $\lim_{n \to \infty} a_n(n)$.