The book of Thoms Jech on set theory mentions some equalities involving $\aleph_{\omega}$:  $\aleph_\omega^{\aleph_1} = \aleph_\omega^{\aleph_0} \cdot 2^{\aleph_1}$ If $\alpha &lt; \omega_1$, then $\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0} \cdot 2^{\aleph_1}$. If $\alpha &lt; \omega_2$, then $\aleph_\alpha^{\aleph_2} = \aleph_\alpha^{\aleph_1} \cdot 2^{\aleph_2}$  How can the above equalities be proved?
So basically I want to show:  $\aleph_{\alpha}^{\aleph_{1}} = \aleph_{\alpha}^{\aleph_{0}}\cdot2^{\aleph_1}$ , Where $\omega\leq\alpha&lt;\omega_1$  There is a proof that I found;    But I don't understand his proof for the limit case of $\alpha$.  To be more specific, I dont understand why $ \left( \sup_{\omega \le \beta &lt; \alpha} \aleph_{\beta}\right)^{\aleph_1} \le \left( \prod_{\omega \le \beta &lt; \alpha} \aleph_{\beta}\right)^{\aleph_1}$  Any insights or help is deeply appreciated.  Cheers
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There is no need to downvote to 'alert' other users that a question isn't worth spending time on - if it's bad enough that it can't be fixed, and so bad that you want to be sure others avoid it then it's not worth keeping open.  These are all situations where the person downvoting should be taking a different action.  , and several questions have thrown ideas at how to increase question upvoting artificially, and a  was even implemented to encourage this behavior.  Given all this, I am now convinced that voting on questions is an unnecessary distraction.  Voting no longer provides any useful information about a question, and there are more than adequate means of fixing or removing questions that need work.  Voting makes more sense for sites like Meta where questions aren't really questions, but suggestions and discussions.  For purely Q/A sites such as SO, SF, SU, the voting is superfluous.  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