Is there any other way to prove this fact? (non-existence of slowest diverging series)

First : $a_n= S_n - S_{n-1}$

Let $U_n$ be : $U_n = \frac{S_n- S_{n-1}}{S_n} = 1 - \frac{S_{n-1}}{S_n}$

Then : $\ln(S_n) -\ln(S_{n-1}) = -\ln(1-U_n)$

Suppose the series of general term $(U_n)$ converges:

$\implies (U_n) \rightarrow 0$

$\implies$ $\ln(S_n) - \ln(S_{n-1}) = U_n + o(U_n) $

It means that $( \ln(S_n) - \ln(S_{n-1}) )$ is the term of a convergent series, thus implying that $(S_n)$ is a converging sequence, which is absurd.

Hence the series of general term $(U_n)$ diverges, always.

Now one interesting question is: what c does make $U_n(c) = \frac{a_n}{S_n^c}$ converge? You know that $c \geq 1$ , but is it the inf?