Find with proofs the infimum, supremum of $E = {a^{n} : n \in \mathbb{N}}$

Question

Find with proofs the infimum, supremum, maximum, minimum of the following set or prove non-existence $$E = \{a^{n} : n \in \mathbb{N}\}, a \in (0,1),$$ a is a fixed number.

Could anyone give me a hint?

Answer 1

There is no minimum because $0<a^n$ for all $n$. But $\lim_{n\to \infty}a^n=0$ so the infimum is $0$. There is a maximum because the sequence above is decreasing: $a^1>a^n$ for all $n$, so, the maximum and the supremum as well are $a$.