Proofs from THE Book - Buffon's Needle Problem (Expectation Function Question) In Proofs from THE Book I have been struggling to understand a certain implication in the proof of Buffon's Needle problem: They introduce the Expectation function E and prove the linearity of it by showing $E(x+y)=E(x)+E(y)$, furthermore they show that $E(rx)=rE(x)$ holds for all rationals $r$. Now, the problem I have understanding: They say since this holds for all rationals and that E is monotone (which is clear), it holds for all real numbers. How does the "monotone" and "hold for rationals" imply that it holds for all of the reals? I suspect it has something to do that the rationals are dense in the reals, but unsure of the "real Mathematical reason". Note: I understand that the Expectation function is known to be linear (though I have very little stats knowledge), but I would like to understand the authors' logic in this scenario.

Given $a\
otin\mathbb Q$, you can form two sequences of rationals, one that converges to $a$ from the below and one that converges from above. Since $E$ is monotonic, that gives you a series of lower bounds and a series of upper bounds for $E(ax)$. Both converge to $aE(x)$, hence we must have $E(ax)=aE(x)$ also for $a\
otin\mathbb Q$. A similar proof should work for the addition law.