Let $C$ be an irreducible, complete smooth curve over an algebraically closed field $k$ (not necessarily equal to $\mathbb C$).
According to Hartshorne Chapter IV, Corollary 3.2 (b), a line bundle $L$ on $C$ is very ample as soon as the degree of $L$ satisfies $deg(L)\geq 2g+1 \quad (\bigstar)$
Since $deg(\omega_C)=2g-2$ , we have $deg (\omega_C^{\otimes n})=n(2g-2)$ and because of $(\bigstar)$, the line bundle $\omega_C^{\otimes n}$ will be very ample as soon as $n(2g-2)\geq 2g+1$.
So for $g=2$ this translates as $n\geq 5/2$, i.e. $n\geq 3$ and for $g\geq 3$ it translates as $n\geq \frac {2g+1}{2g-2}=1+\frac {3}{2g-2}$ i.e. $n\geq 2$.
The inequalities you mentioned that guarantee very ampleness are thus justified.