Proposition 4.7.2.14. Let $(T, \leq )$ be a linearly ordered set and let $\kappa = \mathrm{cf}(T)$ be its cofinality (Definition 4.7.1.28). Then $\kappa $ is a cardinal.
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Proof. Choose a well-ordered set $(S, \leq )$ of order type $\kappa $ and a cofinal function $f: S \rightarrow T$. If $\kappa $ is not a cardinal, then we can choose another well-ordering $\leq '$ of $S$ having order type $\alpha < \kappa $. Applying Proposition 4.7.1.33, we obtain $\mathrm{cf}( T ) \leq \alpha < \kappa $, which is a contradiction. $\square$