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).
![](../../symbols/{x_k}_k.gif)
![](../graphics/incseq.gif)
![](../../symbols/{x_k}_k.gif)
![](../graphics/decseq.gif)
Proof:
Let's look at the first statement, i.e. the sequence in monotone increasing. Take an![](../../symbols/epsi.gif)
![](../../symbols/epsi.gif)
![](../../symbols/epsi.gif)
xk > c -for all k > N, or![]()
| c - xk | <for all 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.