Let $s(n)$ denote the sum of the proper divisors of $n$. For example, $s(6)=1+2+3=6$.
In 1976, Erdős claimed that (top of page 642) for every $k$ and $\delta>0$ and for all $n$ except a sequence of density $0$, it is true that $$(1-\delta)n\left(\frac{s(n)}{n}\right)^i<s^i(n)<(1+\delta)n\left(\frac{s(n)}{n}\right)^i\text{ for every }1\leqslant i\leqslant k.$$ However, only the first inequality was proven.
In 1990, Erdős retracted his claim that (middle of page 169) for every $k$ and $\epsilon>0$ and for all $n$ except a sequence of density $0$, it is true that $$\left|\frac{s(n)}{n}-\frac{s^{j+1}(n)}{s^j(n)}\right|<\epsilon\text{ for every }1\leqslant j\leqslant k.$$
I conclude that these inequalities are supposed to be equivalent, but I cannot show that this is the case. Are these inequalities equivalent?