Prove by cases that for all positive integers $n^n > n^{n-1}$

Question

Prove by cases that for all positive integers $n^n > n^{n-1}$

How am I going to make cases for this proposition? There are an infinite number of positive integers.

Answer 1

$$n^n > n^{n-1} = \frac{n^n}{n} \tag{*}$$

Well we know $n \ne 0$ so $(*)$ is equivalent to

$$1 > \frac{1}{n} \tag{**}$$

$(**)$ is true for $n > 1$

Therefore $(*)$ is true if $n > 1$

Both $(**)$ and $(*)$ do not hold for $n=1$