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  "body": "\n## Statement\n\nLet $A = (a_{nk})$ and $B = (b_{nk})$ be [[regular|29C7C41A52]] [[matrix summability methods|M6311DWJ1E]].\nA sequence $x = (x_k)$ is $A$-summable if each series $(Ax)_n = \\sum_{k} a_{nk} x_k$ converges and the limit $\\lim_A x = \\lim_{n \\to \\infty} (Ax)_n$ exists; the same holds for $B$.\nAssume that every [[bounded sequence|05FQ3VGAR9]] that is $A$-summable is also $B$-summable.\nThen $\\lim_A x = \\lim_B x$ for every bounded $A$-summable sequence $x$.\n\n## Remarks\n\nThe result is also called the bounded consistency theorem: inclusion of the bounded summability fields forces [[consistency|W3BVN1HDXW]] on them.\nIts proof is functional-analytic.\nIt treats the bounded $A$-summable sequences as a space of sequences with the supremum norm.\nOn this space it studies the linear functional $\\lim_B - \\lim_A$, which vanishes on convergent sequences because both methods are regular.\n\nExample: every bounded sequence that is Cesàro summable of order $1$ is Cesàro summable of order $2$.\nThe theorem then says at once that the two Cesàro limits agree on such sequences.\n\nThe conclusion is only about bounded sequences.\nThe theorem says nothing about unbounded sequences in the summability field of $A$.\n\n## Explanations\n\n### L1 human\n\nA matrix method turns a sequence into a new sequence of weighted averages and takes its limit.\nSuppose one such method can handle every bounded sequence that a second method can handle.\nThe theorem says that then the two methods also give the same answer on those sequences.\nSo, for bounded sequences, \"summing more\" never means \"summing differently\".\n\n### L2 human\n\nThe theorem belongs to the functional-analytic side of summability theory, where summability fields are studied as sequence spaces.\nIt complements the [[Silverman–Toeplitz theorem|S4J7SRPXMZ]], which characterizes regular matrices.\nInclusion theorems between classical methods, such as Cesàro and Hölder methods, become consistency theorems on bounded sequences through it.\n",
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