Sequences, series, summability Definition

Subsequence

Definition

Let $(a_n)_{n \in \mathbb{N}}$ be a sequence in a set $X$ . Let $(n_k)_{k \in \mathbb{N}}$ be a strictly increasing sequence of natural numbers: $n_0 < n_1 < n_2 < \cdots$ . The sequence $(a_{n_k})_{k \in \mathbb{N}}$ is a subsequence of $(a_n)$ .

Remarks

Equivalently, a subsequence is the composite $a \circ \varphi$ of the sequence $a \colon \mathbb{N} \to X$ with a strictly increasing map $\varphi \colon \mathbb{N} \to \mathbb{N}$ . By induction, $n_k \ge k$ for each $k$ . The even terms $a_0, a_2, a_4, \dots$ form a subsequence, with $n_k = 2k$ .

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Sequence definition
Subsequence definition

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Tag	`0R8J3ZQMJS`
URI	`m2://0R8J3ZQMJS`
Kind	definition
Writing	W1 stated
MSC	40A05
Defines	Subsequence
Uses	Sequence

Subsequence

Definition

Remarks

Reading path

Where this leads

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