Shortest formal proof that $\displaystyle\sum_{n=1}^{\infty} \left[1 - (a-1)^n\right]$ diverges for all $a\in [0,1]$

Question

Can anyone provide the shortest formal proof that $\displaystyle\sum_{n=1}^{\infty} \left[1 - (a-1)^n\right]$ diverges for all $a\in[0,1]$?

Answer 1

$$\lim_{n\to\infty}\left[1 - (a-1)^n\right]\neq0$$

Answer 2

The $n^\text{th}$ term is $\geqslant 0$ for all $n$, and $\geqslant 1$ for all odd $n$, therefore the $n^\text{th}$ partial sum is $\geqslant n/2$ for all $n$.