Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 19 / 2 / i / b / Solution

König theorem for cardinal numbers states that if ${\kappa }_i<{\lambda }_i$ for each $i\in I$, then ${\sum }_i{\kappa }_i<{\prod }_i{\lambda }_i$. Its cofinality formulation gives, for every infinite cardinal $\kappa$,Taking a cofinal sequence of smaller cardinals in $\kappa$ and comparing their sum with the product of their successors gives this inequality. Another standard consequence, for infinite $\lambda$ and $\kappa \ge 2$, isIndeed, if $\mu =\mathrm{cf}({\kappa }^{\lambda })\le \lambda$, the preceding inequality at ${\kappa }^{\lambda }$ would contradict $({\kappa }^{\lambda }{)}^{\mu }={\kappa }^{\lambda \mu }={\kappa }^{\lambda }$.