I'm wondering how to formally show that $\sqrt{n}o(\sqrt{n}) = o(n)$. The problem I'm having is that I don't really know how to formally resolve the multiplication on the LHS. It would be straightforward to show the result for $\sqrt{n}O(\sqrt{n}) = O(n)$ by simply replacing the $O(\sqrt{n})$ with $c\sqrt{n}$ for some constant $c$.
$f(n) \in o(\sqrt{n})$ only tells me that $\lim_{n\rightarrow\infty}f(n)/\sqrt{n}=0$, which doesn't seem to be very helpful here. Is there an way to treat $o(\sqrt{n})$ analogously to $O(\sqrt{n})$?