Subsequence

analysis

Definition

Subsequence

A subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. ¹

The list of all subsequences for the word “apple” would be “a”, “ap”, “al”, “ae”, “app”, “apl”, “ape”, “ale”, “appl”, “appe”, “aple”, “apple”, “p”, “pp”, “pl”, “pe”, “ppl”, “ppe”, “ple”, “pple”, “l”, “le”, “e”, "" (empty string). ¹

Examples:

$$\begin{array}{rl}(1{)}_{n\ge 0}& \text{ is subsequence of }((-1{)}^n{)}_{n{\ge }_0}\\ (n^2{)}_{n\ge 0}& \text{ is subsequence of }(n{)}_{n{\ge }_0}\end{array}$$

Convergence

Let $(a_n{)}_{n{\ge }_0}$ be a convergent sequence and let $a={\lim }_{n\to \infty }{a}_n=a$, with $a\in \mathbb{R}\cup \{-\infty ,\infty \}$. If $(a_{n_m}{)}_{n{\ge }_0}$ is subsequence of $(a_n{)}_{n\ge 0}$, then:

$$\underset{n\to \infty }{\lim }{a}_{n_m}=a$$

Footnotes

1. Subsequence - Wikipedia ↩ ↩²