Determine the convergence of the sequence $(a_n)$ and find the limit if it exists

Question

Suppose the sequence $(a_n)$ satisfies

$$ 0 < \,\,a_n \,<1,\quad a_{n+1}\,(1-a_n)>\frac{1}{4} \qquad(n=1,2,...). $$
Determine the convergence of the sequence $(a_n)$ and find the limit if it exists.

Answer 1

First, observe that $(a_n-\frac{1}{2})^2\ge 0$. Hence, $a_n^2-a_n+\frac{1}{4}\ge 0$, which rearranges (keeping in mind that $0< a_n< 1$) to $$\frac{1}{4(1-a_n)}\ge a_n$$ Hence, $a_{n+1}>a_n$, so this is a strictly increasing sequence. However, it is bounded above by $1$, so it must converge to a limit. Now, what can that limit $L$ be?