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  "body": "\n## Definition\n\nLet $S$ be a set, and let $f_n \\colon S \\to \\mathbb{R}$ for $n \\ge 1$ and $f \\colon S \\to \\mathbb{R}$ be functions.\nThe sequence $(f_n)$ converges pointwise to $f$ on $S$ if $\\lim_{n \\to \\infty} f_n(x) = f(x)$ for every $x \\in S$.\nThat is, for every $x \\in S$ and every $\\varepsilon \u003e 0$ there is $N$ with $|f_n(x) - f(x)| \u003c \\varepsilon$ for all $n \\ge N$.\n\n## Remarks\n\nHere $N$ may depend on both $\\varepsilon$ and $x$.\nThe pointwise limit is unique, since [[limits|0XKWSSY8P2]] of real sequences are unique.\nA pointwise limit of continuous functions need not be continuous: see [[the sequence $x^n$|0YSRR1JTKZ]]. Mathlib expresses pointwise convergence as convergence in the product topology, through `tendsto_pi_nhds`.\n\n## Explanations\n\n### L1 human\n\nPointwise convergence means that at each single point, the values $f_n(x)$ approach $f(x)$.\nDifferent points may converge at very different speeds.\n",
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