Exercise 3.1.7

Prove $o(g(n)) \cap \omega(g(n))$ is the empty set.

From each term we know that for any positive constant $c > 0$:

$$ \begin{align} \exists & n_1 > 0 : 0 \leq f(n) < cg(n) \\ \exists & n_2 > 0 : 0 \leq cg(n) < f(n) \end{align} $$

If we pick $n_0 = max(n_1, n_2)$, from the problem definition we get:

$$ f(n) < cg(n) < f(n) $$

Which obviously has no solutions. Thus, such function $f(n)$ exists, which means that the intersection is the empty set.