## 3.1. Sequences

### Proposition 3.1.9: Monotone Sequences

If is a monotone
increasing sequence that is bounded above, then the sequence must
converge (see picture).
If is a monotone
decreasing sequence that is bounded below, then the sequence must converge
(see picture).

### Proof:

Let's look at the first statement, i.e. the sequence in monotone increasing. Take an*> 0*and let

*c = sup(x*. Then

_{k})*c*is finite, and given

*> 0*, there exists at least one integer

*N*such that

*x*. Since the sequence is monotone increasing, we then have that

_{N}> c -for allx_{k}> c -

*k > N*, or

for all| c - x_{k}| <

*k > N*. But that means, by definition, that the sequence converges to

*c*.

The proof for the infimum is very similar, and is left as an exercise.