 Solution

The product from the preceding part injects into ${\kappa }^{{\aleph }_0}$, where $\kappa ={\aleph }_{\omega }$, because each coordinate value is an ordinal below $\kappa$. Combining its strict König lower bound with the exponentiation upper bound givesHenceOnly the instance $2^{{\aleph }_{\omega }}={\aleph }_{\omega +1}$ of the generalized continuum hypothesis was needed in these three computations, though the question assumes the full hypothesis. No assumption that countable exponentiation leaves this singular cardinal unchanged is valid: König's theorem rules that out.