For fast-growing $f$, the sum $\sum 1/f(n)$ avoids a given countable set

Let \(S\) be a countable set of real numbers. Show that there are functions \(g_n:\NN\to\NN\) such that if \(f:\NN\to\NN\) satisfies \[ f(n+1)>g_n(f(n))\qquad\text{for all }n, \] then \[ A=\sum_{n=1}^{\infty}\frac1{f(n)} \] converges to a real number that is not in \(S\).