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  "body": "\n## Definition\n\nLet $x = (x_n)_{n \\ge 1}$ be a [[bounded sequence|0A8JNBGT2Z]] of real numbers and let $L$ be a real number.\nThe sequence $x$ is *almost convergent* to $L$ if $\\varphi(x) = L$ for every [[Banach limit|SYF2ND6J97]] $\\varphi$.\n\n## Remarks\n\nExample: every [[convergent sequence|0W4CN9E2W5]] with limit $L$ is almost convergent to $L$, because every Banach limit agrees with the ordinary limit on convergent sequences.\n\nExample: $x = (1, 0, 1, 0, \\dots)$ is almost convergent to $1/2$.\nLet $S$ be the shift and $\\mathbf{1} = (1, 1, \\dots)$.\nThen $x + Sx = \\mathbf{1}$, so for every Banach limit $\\varphi$ one has $\\varphi(x) + \\varphi(Sx) = 1$.\nSince $\\varphi(Sx) = \\varphi(x)$, this gives $\\varphi(x) = 1/2$.\n\nNon-example: let $x_n = 1$ if $4^k \\le n \u003c 2 \\cdot 4^k$ for some integer $k \\ge 0$, and $x_n = 0$ otherwise.\nThis sequence has longer and longer blocks of ones and of zeros, and different Banach limits give it different values.\nThis follows from Lorentz's characterization, since averages over long blocks of consecutive terms can be $0$ or $1$.\n\nThe limit $L$ is unique when it exists, because Banach limits exist.\n",
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