Use of First Principle of Induction
\[n! \gt 6n^2\]
for \[n \ge 6\]
using induction.Let
\[P(k)\]
be the statement that the above statement is true. Then\[6!=720 \gt 216= 6 \times 6"\]
Hence
\[P(6)\]
is true.Suppose that
\[P(k)\]
is true.(\[P(k)\]
implies \[P(k+1)\]
is true.)\[\begin{equation} \begin{aligned} (k+1)! &= (k+1) \times k! \\ & \gt (k+1) \times 6k^2 \\ & \gt (k_1) \times 6(k+1) \; (for \; k \gt 2)\\ &= 6(k+1)^2\end{aligned} \end{equation}\]
Hence
\[P(k+1)\]
is true and the inequality is proved. 
