Exercise 5, § 2

a) Let $\mathbf{f}$ be a regulated function on a bounded open interval ]$a, b$*; suppose that there exists a real function $g$, decreasing on ]$a, b$*, such that $\| \mathbf{f}(x) \| \leqslant g(x)$ on ]$a, b$* and such that the integral $\int_a^b g(t)\,dt$ converges. Show that if $(\varepsilon_n)$ is a sequence of numbers $> 0$, tending to 0, and such that $\inf_{n \geqslant 1} n \varepsilon_n > 0$, one has

b) Give an example of a real regulated function $f > 0$ on ]0, 1] such that the integral $\int_0^1 f(t)\,dt$ converges, and yet the relation (*) does not hold for $\varepsilon_n = 1/n$ (take $f$ so that its value for $x = 2^{-n}$ is $2^{2^n}$).

c) With the same hypotheses as in a) show that