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  "body": "\n## Definition\n\nLet $(a_{m,n})_{m, n \\ge 1}$ be a [[double sequence|B70G8FYPG0]] of real or complex numbers, and let $L$ be a number.\nThe double sequence *converges to $L$ in the sense of Pringsheim* if the following holds.\nFor every $\\varepsilon \u003e 0$ there is an integer $N$ such that $\\abs{a_{m,n} - L} \u003c \\varepsilon$ for all $m \\ge N$ and $n \\ge N$.\nThen $L$ is unique; it is called the *double limit* and is written $\\lim_{m, n \\to \\infty} a_{m,n} = L$.\n\n## Remarks\n\nExample: $a_{m,n} = \\frac{1}{m} + \\frac{1}{n}$ converges to $0$, because $m, n \\ge N$ gives $\\abs{a_{m,n}} \\le \\frac{2}{N}$, which is less than $\\varepsilon$ once $N \u003e 2/\\varepsilon$.\nNon-example: $a_{m,n} = \\frac{m}{m+n}$ has no double limit, because $a_{N,N} = \\frac12$ while $a_{N,n} \\to 0$ as $n \\to \\infty$, so no single $L$ works for $\\varepsilon = \\frac14$.\nA Pringsheim convergent double sequence need not be bounded: put $a_{1,n} = n$ and $a_{m,n} = 0$ for $m \\ge 2$; the double limit is $0$, yet the first row diverges.\nSo the double limit may exist while an iterated limit $\\lim_m \\lim_n a_{m,n}$ does not; the [[Moore–Osgood theorem|QN2210XCST]] gives conditions under which they agree.\n\n## Explanations\n\n### L1 human\n\nBoth indices must be large at the same time.\nIt is not enough to send one index to infinity first and then the other.\n",
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