If $\sup A\not\in A$ then $A$ contains a countably infinite subset

Question

I am required to show that if a bounded non empty set $A\subseteq \mathbf{R}$ is such that $\sup A\not\in A$, then $A$ contains a countably infinites subset.

Now my idea is that for each $k\in\mathbf{N}$ there would exist $a_k\in A$ such that $\sup A-\frac{1}{k}<a_k$ with the required set being $H = \{a_1,a_2,\dots\}$ but how can i modify this construction to ensure that all elements of $H$ are distinct?

Answer 1

Try this: if a set $A\subset\Bbb R$ is finite and nonempty then $\sup A\in A$.