Let S be a bounded non-empty subset of $\mathbb{R}$, and $\bar{m} = sup S$.
133886
Existence of increasing sequence $\{x_n\}\subset S$ with $\lim_{n\to\infty}x_n=\sup S$
55461
How to prove that if $a$ belongs to $\mathbb R$, such that $0\leq a \leq\epsilon$, then $a = 0$
22131
Prove if nonempty $K \subset\mathbb{R}$ is compact, then both sup $K$ and inf $K$ exist and are elements of $K$.
96931
Supremum and infimum of $\{\frac{1}{n}-\frac{1}{m}:m, n \in \mathbb{N}\}$
135694
Proving $\kappa^{\lambda} = |\{X: X \subseteq \kappa, |X|=\lambda\}|$
100565
Property of sup of a set of numbers
48783
Prove that the sum of two compact sets in $\mathbb R^n$ is compact.
78915
Density of the set $S=\{m/2^n| n\in\mathbb{N}, m\in\mathbb{Z}\}$ on $\mathbb{R}$?
110957
No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof?
78463
Prove that $\sup \{-x \mid x \in A\} = -\inf\{x\mid x \in A\}$
106843
Give an example where $\int_{\bar A} f$ exists but $\int_A f$ does not for a continuos $f$ on a bounded open subset $A$ of $\mathbb{R}^n $
18596
A set $A \subset l_1$ is compact if and only if closed, bounded, and one other condition
2575
Showing that $\mathbb{R}$ is connected
153272
Basic question about $\sup_{x\neq 0}{} \frac{\|Ax\|}{\|x\|} = \sup_{\|x\| = 1}{\|Ax\|} $, $x \in\mathbb{R}^n$
81203
Supremum and infimum of sets
143105
Show that, for every real number a, the set {$x ∈ S | f(x) ≤ a$} is a closed subset of $\mathbb R^d$
69184
Proof that absolute value of supremum equals suprememum of absolute value