$\sigma$-additivity of Lebesgue measure. Can one show that Lebesgue measure is $\sigma$-additive on using only its definition which is $\lambda^n ([a,b))= \prod_{i=1}^n(b_i-a_i)$ and the fact that the set of semi-open boxes form a semi-ring? I saw a proof in a book which relies way too heavily on handwavery which isn't particularly helpful as I'm self studying the material.

One defines $\lambda^n$ on intervals as shown, then extends by finite additivity to finite disjoint unions of intervals, which form a semi-ring. To continue the extension, we need to show that result is "countably additive"..., that is, if an element $B$ of the sigma-ring happens to equal a countable disjoint union of other elements $A_k$, then the series converges to the value: $\lambda^n(B) = \sum_k \lambda^n(A_k)$. So first reduce to proving the case where $B$ is an interval and the sets $A_k$ are intervals. Then the proof will proceed (using the Heine-Borel theorem) by reducing to the case of a finite union. And that last case has to be done by some combinatorics.