Let $A$ be a non-empty subset of the real numbers that is bounded below. Let $-A = \{ -x \mid x \in A\}$ Prove that $\text{inf}(A) = -\text{sup}(-A)$
1521
Supremum and Infimum
147627
Relation between condition numbers $\kappa(A^T A)$ and $\kappa(A)$
22131
Prove if nonempty $K \subset\mathbb{R}$ is compact, then both sup $K$ and inf $K$ exist and are elements of $K$.
28822
$|x| - |y| \leq |x-y|?$
92249
Every finite set contains its supremum: proof improvement.
126821
Let $f:[0,1]\to \mathbb{R}$. Show that $\{a \in [0,1]: f(a)\neq 0\} = \bigcup_{n=1}^{\infty} \{a \in [0,1]: |f(a)|\geq 1/n\}$
28007
Assuming: $\forall x \in [0,1]:f(x) > x$ Prove: $\forall x \in [0,1]:f(x) > x + \varepsilon $
147021
Proving a function is bounded
49385
Proving $\forall x (A\to B) \to(A \to \forall x B):x\notin \mbox{free}(A)$ in a Hilbert system where it is not an axiom
77164
If $a \mid m$ and $(a + 1) \mid m$, prove $a(a + 1) | m$.
135694
Proving $\kappa^{\lambda} = |\{X: X \subseteq \kappa, |X|=\lambda\}|$
143130
Proving that the supremum of $(a,b)$ is $b$.
45219
Proving that $C$ is a subset of $f^{-1}[f(C)]$
17461
Show that a set is dense in $[-1,1]$
88982
Prove that for any two sets $A$ and $B$, $A\notin B$ or $B\notin A$
27281
How to prove $\,x^a-1 \mid x^b-1 \iff a\mid b$
123753
In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$