Let $\alpha,\gamma$ be ordinals and $A$ be a set of ordinals. Then $\gamma<\alpha+\sup_{\beta\in A}\beta\implies\exists\beta\in A:\gamma<\alpha+\beta$
99633
Let $α$ be an ordinal and $A$ be a set of ordinals. Then $\sup\limits_{β∈A} (α+β) = α+\sup\limits_{β∈A}(β)$
66398
Union of a finite set and a countably infinite set is countably infinite
45878
Elementary Set Theory: Ordinals, Well Orderings and Isomorphisms
6478
Prove that for any infinite poset there is an infinite subset which is either linearly ordered or antichain.
69499
Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0\alpha$.
3599
show that if ${a_n} \to a$, and $a>0$, then $\exists N$ such that $a_n>0$ for $n \ge N$.
69184
Proof that absolute value of supremum equals suprememum of absolute value
6100
Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$
116527
Validating a proof for necessary and sufficient conditions on sets A & B such that $P(A) \cup P(B) = P(A\cup B)$
138105
Let $f$ and $g$ be continuous at $ x=a$, with $f(a)<g(a)$. Prove that there exists a neighborhood $I$ of a such that $f(x)<g(x)$ for all $x \in I$.
29849
How can I show that $\sup(AB)\geq\sup A\sup B$ for $A,B\subset\mathbb{R}$ where $A\cup B$ is positive and bounded?
40524
Let $A \subseteq (0,\infty$) such that $\inf A>0$. Let $\frac{1}{A}=\{\frac{1}{a},a\in A\}$. Prove that $\sup \frac{1}{A}=\frac{1}{\inf A}$.
131163
There is a well ordering of the class of all finite sequences of ordinals
7112
Product of sets and supremum
43219
If $[K(\alpha):K]=p\neq q=[K(\beta):K]$ then $[K(\alpha+\beta):K]=pq$
146062
A "Cantor-Schroder-Bernstein" theorem for partially-ordered-sets
93771
Prove that for any set $A$ there is some $x \notin A$