Answer
The inequality was proved for $n=1$
The inequality is correct if $n$ is changed by $n+1$
Work Step by Step
Let's prove the inequality for $n=7$:
$(\frac{4}{3})^7\gt7$
$\frac{16384}{2187}\approx7.49\gt7$
It is correct!
Now, suppose that the inequality is correct, that is:
$(\frac{4}{3})^n\gt n$
Now, let's prove the inequality for $n+1$:
$(\frac{4}{3})^{n+1}=(\frac{4}{3})(\frac{4}{3})^n=(1+\frac{1}{3})(\frac{4}{3})^n=(\frac{4}{3})^n+\frac{(\frac{4}{3})^n}{3}$
Since $...(\frac{4}{3})^9\gt(\frac{4}{3})^8\gt(\frac{4}{3})^7=\frac{16384}{2187}\approx7.49\gt3$ we have that:
$\frac{(\frac{4}{3})^n}{3}\gt1$ if $n\geq7$:
$(\frac{4}{3})^{n+1}=(\frac{4}{3})^n+\frac{(\frac{4}{3})^n}{3}\gt(\frac{4}{3})^n+1\gt n+1$
$(\frac{4}{3})^{n+1}\gt n+1$
That is exactly the given inequality if $n$ is changed by $n+1$.
