19 Sequences
The concept of sequences is revisited in the analysis section. As such, this section will relatively be brief while overlapping ideas will only be cross-referenced and a few new ideas will be introduced. The main difference of sequences in this section compared to previous sections is now we deal with the sequence of real numbers.
Monotone and Cauchy Sequences
We have already discussed monotonic sequences and monotone convergence theorem. That is, a monotonic sequence is either decreasing or increasing, denoted
![\[ a_n<a_{n+1} \text{ or } a_n>a_{n+1} \text{ for all } n\in \mathbb{N}\]](https://iu.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-90e5dea931a7459b6ecc49d056c8b51e_l3.png "Rendered by QuickLaTeX.com")
and a bounded monotonic sequence is convergent.
By definition, every convergent sequence approaches to the limit. And thus naturally, the distance between the terms decreases and approaches 0. In fact, the fact that the terms get close to each other is the Cauchy property, and we define Cauchy sequence as follows:
Definition. (Cauchy Sequence)
Cauchy sequence is a sequence 
of real numbers such that
for every 
there exists 
such that
![\[ m,n\geq N\Rightarrow |x_m-x_n|<\epsilon\]](https://iu.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-d9e4f64a2d7351a687e19fa6395a121c_l3.png "Rendered by QuickLaTeX.com")
In other words, Cauchy sequence requires 
to be infinitesimal for arbitrary pairs of large or infinite 
.
Note the similarity with 
–
definition of limit. Therefore, in fact, every convergent real sequence is a Cauchy sequence (Cauchy Convergence Criterion).
Subsequences and Convergence
In fact, we can define a subsequence, a nested sequence of a sequence, or a subset of a sequence. The elements of this subset or subsequence can be picked in an organized way with a pattern, or just arbitrarily. Formally, a subsequence is defined as follows:
Definition. (Subsequence)
Let 
be a sequence, and 
be any monotonically increasing sequence of natural numbers. Then,
![\[ \left\lbrace s_{n_k}\right\rbrace\]](https://iu.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-a3aabd41f99f20ba64e24336a8603c44_l3.png "Rendered by QuickLaTeX.com")
is a subsequence of , where 
.
Note that the index for the subsequence 
is 
, neither 
nor 
.
Let us introduce some useful theorems.
Theorem.
* Every bounded sequence has a convergent subsequence.
* Every unbounded sequence contains a monotone subsequence that has either 
or 
as its limit.
Following is a simple, but interesting example demonstrating the subsequences of a divergent sequence can converge:
![\[ a_n = (-1)^n = \left\lbrace -1,1,-1,1,\cdots\right\rbrace\]](https://iu.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-c54c85c06002330cd09431ca73131f99_l3.png "Rendered by QuickLaTeX.com")
It is clear the original sequence is divergent, since the sequence alternates between -1 and 1, and yet bounded above and below, by 1 and -1, respectively.
If we define subsequences 
and 
for 
, then these subsequences converge and have limits of -1 and 1, respectively.