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 ${\mathrm{\{}{a}_n\mathrm{\}}}_{n\mathrm{\in }\mathrm{N}}$ is a sequence in $X$, then a subsequence of $\mathrm{\{}{a}_n\mathrm{\}}$ is a sequence of the form ${\mathrm{\{}{a}_{n_r}\mathrm{\}}}_{r\mathrm{\in }\mathrm{N}}$ where ${\mathrm{\{}{n}_r\mathrm{\}}}_{r\mathrm{\in }\mathrm{N}}$ 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. ${\{y_n\}}_{n\in \mathbb{N}}$ is a sequence of elements of $X$, and

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

   $$y_n=x_{a(n)}\mathit{\hspace{1em}}\text{ for all }n\in \mathbb{N}.$$

Example.

Let $X=ℝ$ and let $\{x_n\}$ be the sequence

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

Then, the sequence

$${\{y_n\}}_{n\in \mathbb{N}}={\left\{\frac{1}{n^2}\right\}}_{n\in \mathbb{N}}=\{1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\mathrm{\dots }\}$$

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 $a(n)=n^2$.