<s>[INST] How to find the smallest n for inequality involving factorial, $n! > 10^6 $, what is $n$? How to find the smallest n for inequality involving factorial, $n! > 10^6 $, what is n?
I solved it using calculator, that is the answer is $n \approx   9$. How to solve this inequality? 

Smallest $n$ giving $n! > 10^6$ ? 

Edit
inequality relation
 [/INST] $$
\begin{align*}
9!&=1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9\\
&=(2\cdot5)(3\cdot7)(4\cdot6)(8\cdot9)\\
&<10\cdot 30\cdot 30\cdot 80\\
&<8\cdot10^5\\
&<10^6.
\end{align*}
$$
and 
$$
\begin{align*}
10!&=1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10\\
&=(2\cdot5)(3\cdot7)(4\cdot6)(8\cdot9)\cdot10\\
&>10\cdot 20\cdot 20\cdot 70\cdot 10\\
&=28\cdot10^5\\
&>2\cdot10^6\\
&>10^6.
\end{align*}
$$
</s>