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  "body": "\n## Definition\n\nLet $(a_n)_{n \\in \\mathbb{N}}$ be a [[sequence|0QE8STGJH2]] in a set $X$.\nLet $(n_k)_{k \\in \\mathbb{N}}$ be a strictly increasing sequence of natural numbers: $n_0 \u003c n_1 \u003c n_2 \u003c \\cdots$.\nThe sequence $(a_{n_k})_{k \\in \\mathbb{N}}$ is a subsequence of $(a_n)$.\n\n## Remarks\n\nEquivalently, a subsequence is the composite $a \\circ \\varphi$ of the sequence $a \\colon \\mathbb{N} \\to X$ with a strictly increasing map $\\varphi \\colon \\mathbb{N} \\to \\mathbb{N}$.\nBy induction, $n_k \\ge k$ for each $k$.\nThe even terms $a_0, a_2, a_4, \\dots$ form a subsequence, with $n_k = 2k$.\n\n## Explanations\n\n### L1 human\n\nYou get a subsequence when you delete some terms of a sequence and keep infinitely many terms in their old order.\nFrom $1, 2, 3, 4, \\dots$ you can keep the even numbers $2, 4, 6, \\dots$.\n",
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