What follows is not really part of the first section of [Sh:430], but it does pin down an important connection between cov and pp. The proposition is a special case of part of Shelah’s “cov vs. pp theorem”, Theorem 5.4 on page 87 of Cardinal Arithmetic. Note that there are no special assumptions on here. Getting inequalities in the reverse direction (i.e., showing that covering numbers are less than pp numbers) is generally a more difficult proposition.
Let us recall that is the minimum cardinality of a set such that for any (equivalently ), there is a such that .
Proposition 1 If is singular of cofinality , then
Proof: Suppose by way of contradiction that
By the definition of , we can find a (not necessarily increasing) sequence of regular cardinals and an ultrafilter on such that
Let be a family of cardinality standing as witness to , and let be a -increasing and cofinal sequence of functions in . For each , the range of is a subset of of cardinality at most , and so we can find such that
Since is a regular cardinal greater than , there is a single such that
Thus, (by passing to a subsequence of ) we may as well assume that the range of each is a subset of .
But , and so
by our choice of . Let us now define a function by setting if , and
whenever is in . Our assumptions imply that is in .
If , then for each we have
It follows that is a function in such that for all . This is absurd, given our choice of , and the proof is complete.