My textbook, W.E. Gettys, F.J. Keller, M.J. Skove, Physics 1, gives the definition of a reversible transformation as a transformation that can be inverted by effectuating only infinitesimal changes in the surroundings. I admit that I have no idea of what infinitesimal means in a rigourous mathematical language (if we exclude non-standard analysis, but I do not think that it is used in elementary thermodynamics), therefore the definition is quite confusing to me.
Then the book proves that the entropy of a Carnot cycle is zero and, since a reversible cycle can be approximated by the sum of many Carnot cycles, the entropy of a reversible cycle is zero too.
I suppose that we can mathematically formalize the meaning of such an approximation by saying that for any reversible cycle there is a sequence $\{\{C_{1,n},\ldots,C_{n,n}\}\}_n$ of sets of Carnot cycles $C_{1,n},\ldots,C_{n,n}$ (say that the $n$-th set of Carnot cycles used to approximate is composed by $n$ cycles) such that $$\oint\frac{\delta Q}{T}=\lim_{n\to\infty} \sum_{k=1}^n \frac{Q_{C,k,n}}{T_{C,k,n}}+\frac{Q_{H,k,n}}{T_{C,k,n}}.$$As a side note, I realise that, for any Carnot cycle $C_{k,n}$, we have $\frac{Q_{C,k,n}}{T_{C,k,n}}+\frac{Q_{H,k,n}}{T_{C,k,n}}=0$ and therefore, provided that the approximation holds, ${\displaystyle\oint}\frac{\delta Q}{T}=0$.
But, how can we see that a reversible cycle can be arbitrarily approximated by Carnot cycles (i.e. that ${\displaystyle\oint}\frac{\delta Q}{T}=\lim_{n\to\infty} \sum_{k=1}^n \frac{Q_{C,k,n}}{T_{C,k,n}}+\frac{Q_{H,k,n}}{T_{C,k,n}}$, if we use the notation I have introduced and if I have correctly understood the meaning of the approximation)?