= Local uniform Cauchy control for pointwise convergent continuous functions
Suppose <continuous functions> $f_n:\mathbb R\to\mathbb R$ converge <pointwise> to $f$. For each $\epsilon>0$, some nonempty <open interval> $I$ and integer $N$ satisfy $|f_n(x)-f(x)|\leq\epsilon$ for every $x\in I$ and $n\geq N$. Indeed the <closed sets> $Q_N=\bigcap_{n,m\geq N}\{|f_n-f_m|\leq\epsilon\}$ cover $\mathbb R$. The <Baire category theorem> gives an interval inside one $Q_N$, and passing $m\to\infty$ proves the assertion. The interval may depend on $\epsilon$; this does not assert <uniform convergence> on one fixed interval.
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