## 3.1. Sequences

### Examples 3.1.10(b):

Define

The proof is very easy using the theorem on monotone, bounded sequences:
*x*and let_{1}= b*x*for all_{n}= x_{n - 1}/ 2*n > 1*. Then this sequence converges for any number*b*.*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

*L*. Then we have:

But then we have thatlim x_{n}= L = lim x_{n + 1}

so that we have the equation for the unknown limitL = lim x_{n + 1}= lim x_{n}/ 2 = 1/2 lim x_{n}= 1/2 L

*L*:

Therefore, the limit must be zero.L = 1/2 L

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.