An assumption in pcf theory

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 $a=[aleph_2,...,aleph_n,...:n<omega]$ be a set of regular cardinals, say you have a sequence $f_beta$ in $prod a$ of length at most $|a|^+$. Then $sup_beta f_beta in prod a$ since $|a|^+ < min(a)$. But why is this true? If you have for example an $omega_2$ sequence of functions $f:kappa rightarrow kappa$ such that $f(kappa)in kappa$, $kappa$ some $aleph_n$, $n$ not 0 and not 1,then why is $f_beta$ for $beta=omega_2$ outside of the product, as far as we know, we don't know if $2^{aleph_0}= aleph_2 $ since $a$ is a countable set of regular cardinals (say the set of $aleph_n$'s)? Thanks