@TedShifrin @LeakyNun Is the following justification sufficient?

We want to show that the set $\{ 1+a+ \dots+a^k+ \dots+ a^n | n \in \mathbb{N}\}$ is bounded.

We have that $\{ 1+a+ \dots+a^k+ \dots+ a^n | n \in \mathbb{N}\}=\{ \frac{1-a^{n+1}}{1-a} \mid n \in \mathbb{N}\}$ .

Since $|a|<1$, we have that $a^{n+1}>a$. So $1-a^{n+1}<1-a$.

Thus $0<\frac{1-a^{n+1}}{1-a}<1, \forall n$, so we deduce that the set $\{ 1+a+ \dots+a^k+ \dots+ a^n | n \in \mathbb{N}\}$ is bounded. The minimum upper bound is $1$.