- b > 0: the sequence is decreasing and bounded below by 0.
- b < 0: the sequence is increasing and bounded above by 0
- b = 0: the sequence is constantly equal to zero
lim xn = L = lim xn + 1But then we have that
L = lim xn + 1 = lim xn / 2 = 1/2 lim xn = 1/2 Lso that we have the equation for the unknown limit L:
L = 1/2 LTherefore, the limit must be zero.
This proof illustrates the advantage of knowing that a sequence converges. Based on that fact it was easy to determine the actual limit of this recursively defined sequence. On the other hand, it would be very difficult to try to establish convergence based on the original definition of a convergent sequence.