P. Erdős and Leon Alaoglu proved in [**1**] that for every $\epsilon > 0$ the inequality $\phi(\sigma(n)) < \epsilon \cdot n$ holds for every $n \in \mathbb{N}$, except for a set of density $0$.

C. L. mentioned in [**2**] that as a consequence of the previous result one can ascertain that $\displaystyle \lim_{n \to \infty} \frac{\phi(\sigma(n)) }{n} = 0.$

Does anybody know how it is that C. L. proceeded in order to arrive at such a conclusion?

Clearly enough, the fact that an inequality of the type $a_{n} < \epsilon \cdot n$ holds for every $\epsilon > 0$ and a subset of $\mathbb{N}$ of density $1$ does not imply, in general, that the sequence $\displaystyle \frac{a_{n}}{n}$ goes to $0$ as $n \to \infty$.

Hope you guys can shed some light on this inquiry of mine. Let me thank you in advance for your continued support.

**References**

[**1**] L. Alaoglu, P. Erdős: *A conjecture in elementary number theory*, Bull. Amer. Math. Soc. 50 (1944), 881-882.

[**2**] *Mathematical Reflections*, Solutions Dept, Issue #3, 2009, page 23.