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I have researched, convened with classmates, and read a few textbooks over the weekend. I am stumped and have no idea where to start (or finish). I know it pulls from real analysis, but that was so long ago.
2026-04-05 04:48:34.1775364514
$\epsilon-\delta$ proof about integration
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Let $\epsilon >0$.
If $\|f-g\|<\epsilon$, then $\left|\int_0^1f(x)dx -\int_0^1g(x)dx \right| = \left|\int_0^1(f(x)-g(x))dx \right|\leq \int_0^1 |f(x)-g(x)|dx$.
The last quantity is less than $\int_0^1 \|f-g\| dx =\|f-g\|\cdot (1-0) =\|f-g\| < \epsilon$.