We saw that Cost is monoidal closed in Example 2.83
To check if Cost is a quantale, we take an arbitrary set of elements and ask if it has a join.
Because \(\geq\) is a total order, we can take the infimum or greatest lower bound, as the join.
\(\bigvee\{2.5,2.05,2.005,...\} = 2\).
We need a \(0\), which is something which is related to everything (the first join condition is vacuous). Because the preorder relation is \(\geq\) in Cost we need something greater than everything, so \(0 = \infty\).
Thus Cost is a quantale.