Exponential inequality by induction

Base case. $n=4$: $16=2^4\ge4^2=16$. ✓

Inductive step. Assume $2^n\ge n^2$. Then $2^{n+1}=2\cdot2^n\ge2n^2$. It suffices to show $2n^2\ge(n+1)^2=n^2+2n+1$, i.e. $n^2-2n-1\ge0$, i.e. $(n-1)^2\ge2$. True for $n\ge3$.

Conclusion. $2^n\ge n^2$ for all $n\ge4$.