Construct a countably infinite subset A of $\mathbb{R}$ such that sup($A$)$\in A$ and inf($A$)$\notin A$

Question

Let $A=(x_n)$ be a decreasing sequence such that $x_k = 10^{-(k-1)}$ for $k\in \mathbb{N}$. Then sup($A$)$=1 \in A$ and inf($A$)$=0\notin A$.

I'm provided a hint to construct the sequence as a union of two sequences $(a_{2n})$,$(a_{2n+1})$. I could make the evens converge to its supremum and the odds its infimum, but isn't constructing a single sequence as i did above more straightforward?

Answer 1

That is a very good answer.

You can also consider something like the set of rational numbers in the interval $(\sqrt 2,5]$

Answer 2

Another example of such a set is $\{1/n : n = 1,2,3,\dots\}$.