= Solution
{c}
The <Uniform Cauchy criterion> says that $(f_n:E\to\mathbb R)$ <converges uniformly> on $E$ if and only if
$$
\forall\varepsilon>0\ \exists N\ \forall m,n\geq N\ \forall x\in E:\quad |f_n(x)-f_m(x)|<\varepsilon.
$$
For necessity, use the <triangle inequality> through the <uniform limit>. For sufficiency, the condition makes $(f_n(x))_n$ a <Cauchy sequence> for each $x$, so <completeness of the real numbers> defines a <pointwise limit> $f(x)$. Apply the condition with $\varepsilon/2$ and let $m\to\infty$ to obtain $|f_n(x)-f(x)|\leq\varepsilon/2<\varepsilon$ uniformly in $x$.
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