# subsequence

Given a sequence $\{x_{n}\}_{n\in\mathbb{N}}$, any infinite subset of the sequence forms a subsequence. We formalize this as follows:

###### Definition.

If $X$ is a set and $\{a_{n}\}_{n\in\mathbb{N}}$ is a sequence in $X$, then a subsequence of $\{a_{n}\}$ is a sequence of the form $\{a_{n_{r}}\}_{r\in\mathbb{N}}$ where $\{n_{r}\}_{r\in\mathbb{N}}$ is a strictly increasing sequence of natural numbers.

Equivalently, $\{y_{n}\}_{n\in\mathbb{N}}$ is a subsequence of $\{x_{n}\}_{n\in\mathbb{N}}$ if

1. 1.

$\{y_{n}\}_{n\in\mathbb{N}}$ is a sequence of elements of $X$, and

2. 2.

there is a strictly increasing function $a:\mathbb{N}\to\mathbb{N}$ such that

 $y_{n}=x_{a(n)}\quad\text{ for all }n\in\mathbb{N}.$
###### Example.

Let $X=\mathbb{R}$ and let $\{x_{n}\}$ be the sequence

 $\left\{\frac{1}{n}\right\}_{n\in\mathbb{N}}=\left\{1,\frac{1}{2},\frac{1}{3},% \frac{1}{4},\ldots\right\}.$

Then, the sequence

 $\{y_{n}\}_{n\in\mathbb{N}}=\left\{\frac{1}{n^{2}}\right\}_{n\in\mathbb{N}}=% \left\{1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\ldots\right\}$

is a subsequence of $\{x_{n}\}$. The subsequence of natural numbers mentioned in the definition is $\{n^{2}\}_{n\in\mathbb{N}}$ and the function $a:\mathbb{N}\to\mathbb{N}$ mentioned above is $a(n)=n^{2}$.

Title subsequence Subsequence 2013-03-22 12:56:34 2013-03-22 12:56:34 alozano (2414) alozano (2414) 6 alozano (2414) Definition msc 00A05