I have the following statement and i need to demonstrate it: $$f:[a,b]\rightarrow \Bbb R \ \text{is bounded (on } [a,b] \text{) and integrable on} \ [a',b] \ \forall \ a'>a \in \Bbb R \implies f \ \text{is integrable on} \ [a,b]$$
I tried to build a succession of integral when $a' \to a$ but in this kind of question I lack in formalism. The statement is clear and I've fully understood it but i'm not able to put down on paper in mathematical terms my reasoining. The function should be Riemann integrable and I know that probably it could be useful to use the Lebesgue-measure but unfortunately I don't know what it is and for this kind of exercise I'm not supposed to use it. Thank you in advance for your help!
(This was an answer posted in a quite different question. The answer is reposted here, in the hope that it is easier to be found).
To show that, let $\epsilon >0$. Then let $\delta < \epsilon/4C$, where $C$ is the bound of $f$. Since $f$ is integrable on $[a+\delta, b]$, there is a partition $P = \{\delta = x_1< x_2<\cdots< x_n = b\}$ of $[a+\delta, b]$ so that
$$U(P, f|_{[a+\delta, b]}) - L(P, f|_{[a+\delta, b]}) < \epsilon/2.$$
Let $\tilde P$ be the partition of $[a, b]$ given by
$$a = x_0< \delta = x_1 < x_2 <\cdots < x_n = b.$$
Then
$$\begin{split} U(P, f) - L(P, f) &= (M-m) \delta + U(P, f|_{[a+\delta, b]}) - L(P, f|_{[a+\delta, b]})\\ &< 2C\delta + \epsilon /2 \\ &< \epsilon/2 + \epsilon/2 = \epsilon. \end{split} $$
Since $\epsilon$ is arbitrary, $f$ is integrable.
Remark:
One can indeed show that
$$\int_a^b f (x) dx = \lim_{\delta \to 0} \int_{a+\delta} ^b f(x)dx.$$