Can someone give me a hint on this?
Prove: <br />
P(n):\,n^2  < 2^n \, for all
n \in J where n \geqslant 5
Proof by induction
a. P(5):\,\,5^2  < 2^5
25<32 is true;
b. assume that P(k) is true for some <br />
k \geqslant 5 k \in J
c. Show that P(k+1) is true
<br />
\begin{gathered}<br />
  (k + 1)^2  < 2^{k + 1}  \hfill \\<br />
  (k + 1)(k + 1) < 2^k  \cdot 2 \hfill \\<br />
  k^2  + 2k + 1 < 2^k  \cdot (1 + 1) \hfill \\<br />
  k^2  + 2k + 1 < 2^k  + 2^k  \hfill \\ <br />
\end{gathered}
Is the left-hand side less than the right-hand side because the degree of the right is bigger? If so how do I prove that?
Thanks for any hints