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  "body": "\n## Statement\n\nLet $A = (a_{nk})_{n, k \\ge 0}$ be an infinite complex matrix whose [[matrix method|RKFB6AHPBE]] is [[regular|BXVVXJQV8M]].\nThen there is a sequence $s$ with all terms in $\\set{0, 1}$ that is not $A$-summable.\nIn particular, no regular matrix method sums every [[bounded sequence|05FQ3VGAR9]].\n\n## Remarks\n\nProof idea: a \"gliding hump\" argument.\nBy the [[Silverman–Toeplitz theorem|S4J7SRPXMZ]], each fixed column of $A$ fades as $n$ grows, while row sums tend to $1$ and the absolute row sums stay bounded.\nSo one can pick rows $n_1 \u003c n_2 \u003c \\cdots$ and disjoint blocks of columns such that row $n_j$ has almost all its weight on block $j$.\nPut $s_k = 1$ on the blocks with odd $j$ and $s_k = 0$ elsewhere.\nThen $(As)_{n_j}$ is close to $1$ for odd $j$ and close to $0$ for even $j$, so $(As)_n$ does not converge.\nExample: for Cesàro summation, the sequence equal to $1$ for $2^{2j} \\le k \u003c 2^{2j+1}$ and $0$ otherwise has Cesàro means that oscillate between values near $\\frac13$ and near $\\frac23$.\n\n## Explanations\n\n### L1 human\n\nEvery reasonable averaging method of matrix type fails on some sequence of zeros and ones.\nSo there is no single regular matrix method that gives a limit to all bounded sequences.\n\n### L2 human\n\nThe theorem shows that the bounded sequences are never all summable by one regular matrix.\nExtensions of the limit to all bounded sequences do exist, such as [[Banach limits|SYF2ND6J97]], but they are not given by matrices, and their existence uses the Hahn–Banach theorem.\n",
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