Subsequence

A subsequence of a sequence $(a_n)_{n\ge 1}$ is a sequence of the form $(a_{n_k})_{k\ge 1}$, where $n_1<n_2<n_3<\cdots$ is a strictly increasing sequence of indices.

Subsequences are a standard way to isolate “partial” limiting behavior of a sequence; they are used in defining limit superior and limit inferior . The monotone subsequence lemma guarantees monotone subsequences under mild hypotheses.

Examples:

• If $a_n=(-1)^n$, then $(a_{2k})_{k\ge 1}$ is the subsequence $1,1,1,\dots$.
• If $a_n=n$, then $(a_{2k})_{k\ge 1}$ is the subsequence $2,4,6,\dots$.