## 3.1. Sequences

### Examples 3.1.10(a):

First, let us consider the sequence . It is decreasing because:Also, the sequence is bounded below by 0, because each term is positive. Hence, the sequence must converge.( 1/n ) - (1 / (n+1) ) > 0

Note that this does not tell us the actual limit. But we have proved before that this sequence converges to 0.

Next, we consider the sequence . This sequence is increasing because

The sequence is also bounded above by 1, becausen / (n+1) - (n+1) / (n + 2) < 0

*n < n + 1*so that

Hence, the sequence must converge.n / (n + 1) < 1

Note that this does not tell us what the limit of the sequence is. However, the limit is equal to 1, as you can easily prove yourself.