If $f_n$ and $f: E \to \mathbb{R}$ are continuous, E is a bounded set, and $f_n \to f$ pointwise on E, is it true that each function $f_n$ is bounded on E?
2026-02-23 23:05:07.1771887907
$f_n \to f$ pointwise implies $f_n$ is bounded
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Take $f_n(x)=\frac 1 {nx}$ on $E=(0,1)$ and $f(x)=0$.