My approach to solving this: By induction.
(1) $S(n) = (n^2 < 2^n)$ for all $n \geq 6$, $n \in \mathbb N$.
(2) Base Case: $n = 6$
$$6^2 < 2^6$$ $$36 < 64$$
So the statement is true for $n = 6$. Thus it has been shown that $S(6)$ holds.
(3) Induction Hypothesis: Assume $S(n)$ holds for some $n$.
$$S(n) = (n^2 < 2^n)$$
(4) Inductive Step: Show that $S(n+1)$ holds.
$$\begin{align} (n+1)^2 &< 2^{n+1} \\ n^2 + 2n + 1 &< 2^{n+1} \\ n^2 + 2n + 1 &< 2 \cdot 2^{n} \end{align}$$
At this point I have no idea how to proceed. Could anyone give me a hint on how to proceed with what I have so far?