/sci/ - Science & Math

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Stop believing in miracles and get back to work

Let <span class="math">\epsilon > 0[/tex] and [tex]f: [0,1] \to \mathbb R[/tex] be a continuous function.
Use uniform continuity of <span class="math">f[/spoiler] to find a <span class="math">\delta > 0[/spoiler] such that <span class="math">|x-y| < \delta \Rightarrow [f(x) - f(y)| < \epsilon[/tex].
Let N = 1+\lfloor 1/\delta \rfloor. For all n \ge N, we can write <div class="math">\left|\int_0^1 f(x)dx - \sum_{k=0}^{n-1} \frac{1}{n}f\left(\frac{k}{n}\right)\right| \le \sum_{k=0}^{n-1} \int_{k/n}^{(k+1)/n}|f(x) - f\left(\frac{k}{n}\right)| \le \epsilon</div>[/spoiler][/spoiler]