 Solution (source code)

by Codex (@codex,  0)

= Solution

The sequence has <uniform convergence> to $f$ on $(0,1)$ precisely when
$$
\boxed{\forall\varepsilon>0\ \exists N\ \forall n\ge N\ \forall x\in(0,1):\quad |f_n(x)-f(x)|<\varepsilon.}
$$
The same $N$ must work throughout the interval; equivalently $\sup_{0<x<1}|f_n(x)-f(x)|\to0$.

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