Are there any subsets $A, B$ of $\mathbb R$ such that $\sup(B)>\sup(A)$ but none of the elements of $B$ are upperbounds of $A$?
50142
If sup A $\lt$ sup B show that an element of $B$ is an upper bound of $A$
67259
Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?
1307
Infimum and supremum of the empty set
90566
There exists surjections from A onto B and vice versa, and there is no bijection between the sets?
125043
Show that F(A−B) is not necessarily a subset of F(A)−F(B).
92249
Every finite set contains its supremum: proof improvement.
133886
Existence of increasing sequence $\{x_n\}\subset S$ with $\lim_{n\to\infty}x_n=\sup S$
138283
$|(a,b)| = |\Bbb R|$ ? Cardinality of any open interval
27899
Can we prove the existence of $A\cup B$ without the union axiom?
7974
lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $
858
Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$?
20706
Non-existence of a Surjective Function from a Set to Its Subsets (Cantor's theorem)
74330
$\mathbb{R}^n$ not homeomorphic to open subset of $\mathbb{R}^m$
98391
example of a nonempty subset is closed under scalar multiplication but not a subspace
136106
Is every open subset of $ \mathbb{R} $ uncountable?
45405
An elementary way to show any bounded subset of $\Bbb{R}^k$ is totally bounded
11405
If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?