my question is as the title says. I've tried proving by induction but got stuck: for $n=1$, $$a^1\le a$$ now assumimg that $$a^n \le a$$ for every $$ a \in \mathbb{R}, 0<a<1, n \in \mathbb{N}$$ now for the induction step: prove that $$ a^{n+1} \le a $$ $$a^n * a \le a$$ since $$ 0<a<1$$ and more specifically $$a \neq 0$$ we can divide by a, such that $$ a^n \le 1 $$ using the assumption, we know that $$a^n \le a <1 $$ and therefore $$a^n < 1$$ but I couldn't understand how to prove that $$ a^n \le 1 $$
Sorry for any possible mistakes, I've never posted before and English is not my mother language. Thanks a lot.
