Limit of Integer to Reciprocal Power/Proof 3

Proof
Let $n^{1/n} = 1 + a_n$.

The strategy is to:
 * $(1): \quad$ prove that $a_n > 0$ for $n > 1$
 * $(2): \quad$ deduce that $n - 1 \ge \dfrac {n \paren {n - 1} } {2!} a_n^2$ for $n > 1$

and hence:
 * $(3): \quad$ deduce that $0 \le a_n^2 \le \dfrac 2 n$

Let $n > 1$.

Then: