I know that the sequence of functions converges pointwise to the following piece-wise function $$ f(x) = \begin{cases} \frac{1}{x}, & 0<x\leq 1 \\ 0, & x = 0 \end{cases} $$ but I suspect that it does not display uniform convergence.
2026-04-24 11:26:20.1777029980
On $[0,1]$ does $f_n = \frac{nx}{1+nx^2}$ converge uniformly?
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Take $x=1/n$. Then
$$|f_n(x)-f(x)| = \Big|\frac{n}{n+1} - n\Big|$$
does not converge, so there is no uniform convergence.