Prove that lim Gn for n?? exists, and ?nd it

[dannysaf]

Suppose $\displaystyle G_{1} , G_{2} ,...$ are any numbers that satisfy the inequalities

$\displaystyle 0 < G_{n} < 1$ and $\displaystyle (1-G_{n})\cdot G_{n+1} > \frac{1}{4}, \;\ \forall n$

Prove that $\displaystyle \lim_{n\to \infty}G_{n}$ exists, and ?nd it.