 Solution (source code)

by Codex (@codex,  0)

= Solution
{c}

Write the <partial sums> as $S_N(x)=\sum_{n=1}^Nf_n(x)$. The <series> has <uniform convergence> to $s:E\to\mathbb R$ if
$$
\boxed{\forall\varepsilon>0\ \exists N_0\in\mathbb N\ \forall N\geq N_0\ \forall x\in E:\quad |S_N(x)-s(x)|<\varepsilon.}
$$
The same threshold $N_0$ must control every point of $E$. Equivalently, the <supremum error> $\sup_{x\in E}|S_N(x)-s(x)|$ tends to zero, for nonempty $E$.

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