# Commuting limits in relating the harmonic series to coprimality densities

Let $f(b, q)$ be the reciprocal proportion of naturals less than $b$ which are not divisible by any prime less than $q$. Note that $\displaystyle \lim_{q \to \infty} f(b, q) = b$, while $\displaystyle \lim_{b \to \infty} f(b, q)$ is the sum of the reciprocals of those positive integers with prime factors all less than $q$. Thus, $\displaystyle \lim_{b \to \infty} \lim_{q \to \infty} f(b, q) = \infty$, while $\displaystyle \lim_{q \to \infty} \lim_{b \to \infty} f(b, q)$ is the harmonic series.

Given that we already know the harmonic series diverges, we may conclude that these limits commute, but is there any way to reason in the opposite order? That is, is there any good reason that these limits should commute, such that we can conclude by this reasoning that the harmonic series must diverge?

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