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To prove the Archimedean property, suppose the positive integers were bounded above in $\mathbb{R}$. By the least upper bound axiom they would have a finite supremum $s$. Since $s-1$ is not an upper bound, some positive integer $k$ satisfies $k>s-1$. Then the positive integer $k+1>s$, contradicting the definition of $s$. Thus the positive integers are unbounded. Equivalently, for any positive real $a,b$, some positive integer $n$ satisfies $na>b$, by applying unboundedness to $b/a$.

For fixed $m$, put $r_m={\cos }^2(m!\pi x)$. The standard properties needed are $0\le {\cos }^{2}t\le 1$, and ${\cos }^{2}t=1$ exactly when $t\in \pi \mathbb{Z}$. If $0\le r<1$, then $r^n\to 0$; for $0<r<1$ this also follows from the Bernoulli inequality, since $r^{-n}\ge 1+n(r^{-1}-1)\to \infty$. Therefore the inner limit isIf $x=p/q$ is a rational number, with $q\ge 1$, then $q$ divides the factorial $m!$ whenever $m\ge q$. Hence $m!x$ is an integer for all sufficiently large $m$, and the outer limit is one. This includes the endpoints $x=0,1$. If $x$ is an irrational number, $m!x$ cannot be an integer for any $m$, since division by $m!$ would make $x$ rational. Then every inner limit is zero. The factorial cosine iterated limit detects rationality result is consequentlyEach inner limit was taken with $m$ fixed; the argument does not interchange the two limits or assert a joint limit as both indices grow.