Proposition 1 (Cauchy’s criterion for integrals)

Let $f$ be a regulated function on an interval $I \subset \mathbf{R}$ having endpoints $a$ and $b$ ($a < b$). For the integral $\int_a^b f(t) \, dt$ to exist it is necessary and sufficient that for every $\varepsilon > 0$ there exist a compact interval $J_0 = [\alpha, \beta]$ contained in $I$, such that for any compact interval $K = [x, y]$ contained in $I$ and having no interior points in common with $J_0$, one has $\| \int_K f(t) \, dt \| \leq \varepsilon$.

Indeed, since $E$ is complete the Cauchy criterion shows that for the integral $\int_I f(t) \, dt$ to be convergent it is necessary and sufficient that for any $\varepsilon > 0$ there exists a compact interval $J_0 = [\alpha, \beta]$ with for every compact interval $J$ such that $J_0 \subset J \subset I$ one has $\| \int_J f(t) \, dt - \int_{J_0} f(t) \, dt \| \leq \varepsilon$. The proposition will follow from the following lemma: