PlanetMath: subsequence

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subsequence

(Definition)

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

Definition 1 If is a set and $\{a_n\}_{n \in \mathbb{N}}$ is a sequence in , 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

$\{y_n\}_{n\in\mathbb{N}}$ is a sequence of elements of , and
there is a strictly increasing function $a:\mathbb{N}\to \mathbb{N}$ such that
$\displaystyle y_n = x_{a(n)}$ for all $\displaystyle n\in\mathbb{N}.$

Example 1 Let $X=\mathbb{R}$ and let $\{x_n\}$ be the sequence

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

Then, the sequence

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

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

"subsequence" is owned by alozano. [ full author list (2) | owner history (1) ]

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Cross-references: function, natural numbers, strictly increasing, infinite subset, sequence
There are 37 references to this entry.

This is version 3 of subsequence, born on 2002-08-16, modified 2007-06-22.
Object id is 3300, canonical name is Subsequence.
Accessed 5032 times total.

Classification:

AMS MSC:

00A05 (General :: General and miscellaneous specific topics :: General mathematics)

Pending Errata and Addenda

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