Proving a Dedekind Cut for Rational Numbers

Question

For Dedekind cuts α, β > $0^*$, I need show that α ⊘ β := {p ∈ Q | p · s < r for some r ∈ α, s ∈ $β^c$ such that r,s > 0 and s not the lowest element.

I know the properties of Dedekind Cuts are:

1. α is non-empty (and a proper subset);
2. if p ∈ α and q < p, then q ∈ α;
3. α has no largest element, i.e. q ∈ α such that p ≤ q for all p ∈ α.

I was able to prove the first 2 properties but not the third.

How can I start?