I have seen this claim on Winston Ou Real Analysis Lecture on youtube but he does not really explain how to get this conclusion
$f_n = \chi_{[n,n +1)}$ converges pointwise to 0.
2026-02-26 10:52:16.1772103136
Characteristic function converges pointwise to 0 - Baby Rudin
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$f_n(x) = 0$ if $n > x$, so the limit as $n \to \infty$ is $0$.