User blog comment:Ubersketch/Fundamental sequences for CK and higher?/@comment-35295276-20181025210550/@comment-35470197-20181026235625

Let \(\alpha\) be a countable limit ordinal. Take a bijective map \(\iota \colon \omega \to \alpha\), which exists by the assumption of the countability of \(\alpha\). Put \(m_0 := 0\) and \(m_{n+1} :=\min \{m \in \omega \mid \iota(m) > \iota(m_n)\}\). Then you get a fundamental sequence \((\iota(m_n))_{n \in \omega}\).