Often in theorems of pcf theory one has the assumption that the length of sequences of functions has to respect some bound so things can be well-defined. For instance, let a=[aleph2,...,alephn,...:n<omega] be a set of regular cardinals, say you have a sequence fbeta in proda of length at most |a|+. Then supbetafbetainproda since |a|+<min(a). But why is this true? If you have for example an omega2 sequence of functions f:kapparightarrowkappa such that f(kappa)inkappa, kappa some alephn, n not 0 and not 1,then why is fbeta for beta=omega2 outside of the product, as far as we know, we don't know if 2aleph0=aleph2 since a is a countable set of regular cardinals (say the set of alephn's)? Thanks
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