Here one of Koenig's lemmas helps:
Consider two families of cardinals indexed by such that for any then:
Use this result with the fact that given a cardinal
is the lowest cardinal such that there exists a family with for all and
Show that for each infinite cardinal , .
Notation: denotes the cofinality. I know some properties of . They may be helpful.
For each infinite cardinal number , .
However, I do not see how to prove this. Any hints would be great. Thanks.