I am working through a paper and I crossed a point where an inequality becomes an equality, and I am not sure why does it happen. The thing is that one of the term is substituted by a small-o notation term, and so I was wondering if the following is true, taking into account that $f$ is positive:
$f\leq o(g)\rightarrow f=o(g)$
I don't think this is true without further assumptions. For instance take $f(x)=-\frac{1}{x}$, $g(x)=x$ and $h(x)=x^2$, say defined on $(0,1)$. You have $h=o(g)$ for $x\rightarrow0,$ $f(x)\leq h(x)$ but $f$ is not $o(g)$ for $x\rightarrow0$. Indeed $\frac{f}{g}=-\frac{1}{x^2}\rightarrow-\infty$ as $x$ goes to $0$. But if you assume $f$ to be positive then it is true (by the sandwich theorem).