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  "body": "\n## Summary\n\nPringsheim convergence is the usual notion of convergence for a [[double sequence|B70G8FYPG0]] $(a_{m,n})$.\nThe double sequence converges to $L$ when $a_{m,n}$ is close to $L$ as soon as both indices $m$ and $n$ are large.\nIt is the basis for the theory of double series and for comparing double limits with iterated limits.\n\n## Explanations\n\n### L1 human\n\nThink of the terms $a_{m,n}$ written in an infinite table with rows $m$ and columns $n$.\nPringsheim convergence to $L$ means that, outside a large enough corner square of the table, every entry is within any given tolerance of $L$.\nFor example, $a_{m,n} = \\frac{1}{m} + \\frac{1}{n}$ tends to $0$ in this sense.\nOn the other hand, $a_{m,n} = \\frac{m}{m+n}$ has no double limit: on the diagonal it is $\\frac12$, but far out along a row it is near $0$.\n",
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