query: For any $a\in S_\Omega$, either it has an immediate predecessor or there is an increasing sequence in $S_\Omega$ whose l.u.b. is $a$.
positive_passages.docid: 148927
positive_passages.text: Why does every countable limit ordinal have cofinality $\omega$?
negative_passages.docid: 111449
negative_passages.text: Prove there is an increasing sequence ($s_n$) of points in $S$ such that $\lim s_n = \sup S$.
negative_passages.docid: 42300
negative_passages.text: Let the sequence be defined recursively $X_{n+1}=pX_{n}+q$ for p nonzero and $X_{1}$ arbitrary. In what conditions does the sequence converge?
negative_passages.docid: 34298
negative_passages.text: sequentially continuous on a non first-countable
negative_passages.docid: 53005
negative_passages.text: Properties of $\liminf$ and $\limsup$ of sum of sequences: $\limsup s_n + \liminf t_n \leq \limsup (s_n + t_n) \leq \limsup s_n + \limsup t_n$
negative_passages.docid: 149822
negative_passages.text: If $\omega \in \Omega^q(M)$, and $(\omega|_{U})_p = 0$, is $\omega_p = 0$?
negative_passages.docid: 99633
negative_passages.text: Let $α$ be an ordinal and $A$ be a set of ordinals. Then $\sup\limits_{β∈A} (α+β) = α+\sup\limits_{β∈A}(β)$
negative_passages.docid: 45939
negative_passages.text: For which $n≥2$ does there exist $\sigma\in A_n$ with $|\sigma| > n$
negative_passages.docid: 72654
negative_passages.text: If $(a_n)$ is a null sequence and $(b_n)$ is bounded, then $(a_nb_n)$ is a null sequence
negative_passages.docid: 143545
negative_passages.text: If $(a_n)$ is Cauchy it has a subsequence $(a_{n_i})$ such that $d(a_{n_{i+1}},a_{n_i})<2^{-i}$ for all $i$.
negative_passages.docid: 141895
negative_passages.text: Proving that a sequence is monotone
negative_passages.docid: 46778
negative_passages.text: Prove that a recursive sequence is monotonically increasing
negative_passages.docid: 53363
negative_passages.text: Ultralimit of an eventually constant generalized sequence
negative_passages.docid: 84900
negative_passages.text: Concerning the sequence $\Big(\dfrac {\tan n}{n}\Big) $
negative_passages.docid: 79767
negative_passages.text: Properties of $x_k=\frac{a_{k+1}-a_{k}}{a_{k+1}}$ where $\{a_n\}$ is unbounded, strictly increasing sequence of positive reals
negative_passages.docid: 133870
negative_passages.text: if $\alpha\leq\beta$ then there is $\gamma$ such as $\alpha+\gamma=\beta$
negative_passages.docid: 14489
negative_passages.text: Showing that a sequence is monotone and thus convergent