In my real analysis textbook the improper integrals are defined as follows.
Let $a,b \in \mathbb{R}, a < b.$ Let $f$ be defined over the interval of $[a, b)$. For every $\beta \in (a, b),$ let $f$ be Riemman integrable on $[a, \beta]$ and unbound on $(\beta, b)$. We define $$\int_{a}^{b}{f(x)dx} \text{ as an improper integral of the function } f \text{ on } [a, b)$$ where: $$\int_{a}^{b}{f(x)dx} = \lim_{\beta \to b^-}\int_{a}^{\beta}f(x)dx$$
The analogous definition is given for the lower limit.
Also the convergence of the integral is, naturally, defined as the existence of the given limit (and it being finite). Then out of the blue comes the following theorem:
$$\int_{a}^{b}f(x)dx \quad \text{with the singularity at $x=b$ converges if and only if }\quad (\forall \epsilon>0)(\exists\beta\in(a, b)) \text{ such that } (\forall\beta', \beta''\in \mathbb{R}) \text{ if } \beta<\beta' <\beta'' <b \Rightarrow \bigg|\int_{\beta'}^{\beta''}{f(x)dx} \bigg| < \epsilon$$
And the given proof is extremely vague and not understandable. It just says this:
Let $F(\beta)=\int_{a}^{b}f(x)dx$ .The integral $\int_{a}^{b}f(x)dx$ converges if there exists a finite limit $\lim_{\beta \to b^-}{F(\beta)}$. $\bigg| F(\beta'') - F(\beta') \bigg| = \bigg| \int_{\beta'}^{\beta''}f(x)dx \bigg| < \epsilon$ . Proof completed. Though there is a side note saying 'Cauchy's critearia for the existence of a finite limit.
The side note didn't help much as well. Could anyone explain this proof?
Thanks.
Cauchy's criterion for the existence of a limit of a function: Given a function $f\colon[a,b)\longrightarrow\mathbb{R}$, the limit $\lim_{x\to b}f(x)$ exists (in $\mathbb R$) if and only if$$(\forall\varepsilon>0)(\exists\delta>0)\bigl(\forall x,y\in[a,b)\bigr):x,y\in(b-\delta,b)\cap[a,b)\implies\bigl\lvert f(x)-f(y)\bigr\rvert<\varepsilon.$$I suppose that you will agree that this is exactly what is need in the situation that you mentioned.