Suppose $\displaystyle S = (a,b) $ with $\displaystyle 0 < b < a $. We generate points as follows: $\displaystyle x_0 = a, y_0 = b $, $\displaystyle x_{n+1} = \frac{x_n+y_n}{2} $ and $\displaystyle y_{n+1} = \frac{2x_{n}y_{n}}{x_n+y_n} $.

So $\displaystyle 0 < x_{n+1}-y_{n+1} = \frac{x_n-y_n}{x_n+y_n} \cdot \frac{x_n-y_n}{2} < \frac{x_n-y_n}{2} $.

Does the $\displaystyle \frac{x_n-y_n}{2} $ act similarly to $\displaystyle \epsilon/2 $? And so $\displaystyle \lim x_n = \lim y_n = x $?