Subsequence

analysis

Definition

Subsequence

A subsequence of a given sequence $(a_n{)}_{n\in \mathbb{N}}$ is a sequence $(a_{n_k}{)}_{k\in \mathbb{N}}$ formed by selecting elements from the original sequence while preserving their relative order.

Formally, let $n_0<n_1<n_2<\dots$ be a strictly increasing sequence of natural numbers. Then the sequence $(b_k{)}_{k\ge 0}$ defined by $b_k=a_{n_k}$ is a subsequence of $(a_n)$.

Convergence

If a sequence $(a_n{)}_{n\ge 0}$ is a convergent sequence with limit $L$, then every subsequence of $(a_n)$ also converges to $L$:

$$\underset{n\to \infty }{\lim }{a}_n=L⟹\underset{k\to \infty }{\lim }{a}_{n_k}=L$$

Conversely, if a sequence has two subsequences that converge to different limits, the original sequence is divergent.

Properties

• Bolzano–Weierstrass Theorem: Every bounded sequence in ${\mathbb{R}}^n$ has a convergent subsequence.
• Limit Points: A value $L$ is a limit point of a sequence if and only if there exists a subsequence that converges to $L$.

Examples

• Alternating Sequence: For $a_n=(-1{)}^n$, the subsequence of even indices $a_{2k}=1$ converges to 1, while the subsequence of odd indices $a_{2k+1}=-1$ converges to $-1$.
• Increasing Indices: For $a_n=n$, the sequence of squares $a_{k^2}=k^2$ is a subsequence.