Is this a valid proof?
Definition: $f\left(n\right)=\omega\left(g\left(n\right)\right)$ if $\underset{n\rightarrow\infty}{\lim}\frac{f\left(n\right)}{g\left(n\right)}=\infty$ for $f\left(n\right),g\left(n\right):\mathbb N\rightarrow\mathbb R^{+}$
Therefore: $\underset{n\rightarrow\infty}{\lim}\frac{2^{n}}{n}\stackrel{L'H\hat{o}pital}{=}\underset{n\rightarrow\infty}{\lim}\frac{2^{n}\ln\left(2\right)}{1}=\underset{n\rightarrow\infty}{\lim}2^{n}\ln\left(2\right)=\infty$