subsequence
A subsequence is what you get by reading a sequence selectively — keeping some terms, skipping others, but never going backward. Think of skimming a long book and reading only chapters 1, 4, 5, 9, ... in that order: you preserve the original order and pick out an infinite run of terms.
Formally, given a sequence (a_n), choose indices n_1 < n_2 < n_3 < ... strictly increasing; the resulting sequence (a_{n_k}) is a subsequence. The strictly increasing requirement is what forbids reusing or reordering terms, and it forces n_k >= k, so a subsequence always marches off to infinity in index just like its parent.
Subsequences are the lens through which divergence and limit behavior come into focus. A fundamental fact: if a_n -> L, then every subsequence also converges to L. The contrapositive is a powerful divergence test — if two subsequences tend to different limits, the original sequence cannot converge. Values that arise as limits of subsequences are called subsequential limits, and they feed directly into the notions of limit superior and limit inferior.
From a_n = (-1)^n, the even-index subsequence a_{2k} = 1 converges to 1, while the odd-index subsequence a_{2k-1} = -1 converges to -1. Two different subsequential limits prove the parent sequence diverges.
Different subsequential limits are a clean certificate of divergence.
A subsequence must keep infinitely many terms; you cannot stop after finitely many. The condition n_1 < n_2 < ... is over an infinite list of indices.