Hello guys. A moderator on a math forum pointed out errors in some stuff. But I can't quite see what he means, and he's not going to answer me anymore, probably. First, he said that both @Balarka's and my proof of this lemma :
If $\displaystyle \lim_{x \to \infty} \prod_{n\le x} f(n)$ and $g(m) \sim h(m)$, then $$\displaystyle \lim_{m\to \infty} \frac{\prod_{n\le g(m)} f(n)}{\prod_{n<h(m)} f(n)}=1 $$
are wrong, and retain the same error. More precisely, he said the lemma itself is incorrect, namely I need more conditions to reache the desired conclusion.