What is the difference between $O$ (big oh) and $o$ (small oh) notations in asymptotic analysis? Even though I understand that $o$ is used for a bound that is not tight, is it allowed to use $O$ notaion for a bound that is also not tight? For example can I say that $5n=O(n^3)$?
I am also confused by what this statement in CLSR means "The main difference is that in $f(n) =O(g(n))$, the bound $0 \le f(n) \le cg(n)$ holds for some constant $c > 0$, but in $f(n) = o(g(n))$, the bound $0 \le f(n) < cg(n)$ holds for all constants $c > 0$." As the value of $c$ will differ in the inequality "$0 \le f(n)< cg(n)$ for all $n > n_0,c>0$. " for different $n_0$.