detecting cofinality in ordinals

detecting cofinality in ordinals

Suppose K is an unbounded well ordered set, and suppose K is minimal in
the sense that each proper initial segment of K has strictly smaller
cardinality.
Suppose J is an unbounded well ordered set of the same cardinality as K.
Must there exist a subset M of K, and an order preserving embedding
f:M--->J so that f(M) is cofinal in J?
( i.e. for each j in J there exists m in M so that j is less than f(m)).
( In earlier versions of this question, if we overreach and demand that
M=K, Brian and Asaf observed the answer is no, even if each final segment
of J has the same cardinality as K)