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  "body": "\n## Definition\n\nLet $S$ be a set and let $(x_n)_{n \\ge 0}$ be a [[sequence|0G9EKM0E15]] of elements of $S$.\nThe sequence is *periodic* if there is an integer $p \\ge 1$ such that\n$$x_{n+p} = x_n \\quad \\text{for all } n \\ge 0.$$\nSuch a $p$ is a *period* of the sequence, and the smallest such $p$ is its *least period*.\n\n## Remarks\n\nExample: $x_n = (-1)^n$ in $\\R$ satisfies $x_{n+2} = (-1)^n (-1)^2 = x_n$ for all $n$, so it is periodic, with least period $2$ since $x_1 \\ne x_0$.\n\nNon-example: $x_n = n$ is not periodic, since $x_{n+p} = n + p \\ne x_n$ for every $p \\ge 1$.\n\nThe sequence is *eventually periodic* if the condition holds for all $n \\ge n_0$ for some $n_0$; for example $0, 1, 1, 1, \\dots$ is eventually periodic but not periodic.\nEvery period is a multiple of the least period.\nThe same definition applies to two-sided sequences $(x_n)_{n \\in \\Z}$, with the condition required for all $n \\in \\Z$.\n",
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