Remark 4.7.1.29. For any linearly ordered set $(T, \leq )$, the identity map $\operatorname{id}: T \rightarrow T$ is cofinal. Consequently, if $T$ is well-ordered set of order type $\alpha $, then we have $\mathrm{cf}(\alpha ) = \mathrm{cf}(T) \leq \alpha $. Beware that the inequality is often strict.
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