# Question about cofinality of an ordinal

Let $\alpha$ an ordinal and $\langle\alpha_\xi\rangle$ a cofinal sequence of elements of $\alpha$. The length, $\gamma$, of this sequence is at least $\operatorname{cf}\alpha$ but can be equal to any ordinal $\geq \operatorname{cf}\alpha$. If this sequence is a strictly increasing one then $\gamma$ is $\leq$ to the order type of $\alpha$ that is $\alpha$. Is it exact ?

-

If $\alpha$ is a limit ordinal and $\langle\alpha_\xi:\xi<\gamma\rangle$ is a non-decreasing sequence cofinal in $\alpha$, then $\operatorname{cf}\gamma=\operatorname{cf}\alpha$, even if the sequence is not strictly increasing. As you say, if the sequence is strictly increasing, then $\gamma\le\alpha$.