Prove that lim gn for n→∞ exists, and find it.

[dannysaf]

Suppose g1 , g2 ,... are any numbers that satisfy the inequalities
0 < gn < 1 and (1 − gn )gn+1 > 1/4 for all n.

Prove that lim gn for n→∞ exists, and find it.


I need well substantiated answer! Thanks.

 

[LCKurtz]

Suppose g1 , g2 ,... are any numbers that satisfy the inequalities
0 < gn < 1 and (1 − gn )gn+1 > 1/4 for all n.

Prove that lim gn for n→∞ exists, and find it.


I need well substantiated answer! Thanks.

Is this a homework problem? I'll give you a hint. Show:

$$\frac 1 {4(1-x)} \ge x$$

on the interval, consequently showing your sequence g_n is increasing. Then...