Why $\|S\|=\sup_{\|x\| \leq 1}\|Sx\|?$
62425
Equivalent Definitions of the Operator Norm
28822
$|x| - |y| \leq |x-y|?$
7974
lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $
79307
How to analyze $\sup_{x>0}|e^xf(x)| < \infty$ and $\sup_{n\in\mathbb{N}} |f^{(n)}(0)|< \infty$?
86518
Does $X$ have a lower bound?
27482
How to show $-\sup(-A)=\inf(A)$?
126165
Prove $|\arctan x - \arctan y|\leq |x-y|$
12523
Supremum of a Continuous Function is Continuous
63516
$C^1[0,1]$ is not complete with respect to sup norm
82097
If $A \succeq B$, is it true that $B^{-1} \succeq A^{-1}$?
117640
Prove that $\lim \inf \dfrac{x_{n+1}}{x_n}\leq \lim \inf x_n^{1\over n} \leq \lim \sup x_n^{1\over n} \leq \lim \sup \dfrac{x_{n+1}}{x_n}. $
19823
Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$
61799
Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
57055
Are $|X|$ and $\operatorname{sgn}(X)$ independent?
111449
Prove there is an increasing sequence ($s_n$) of points in $S$ such that $\lim s_n = \sup S$.
91877
Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
143130
Proving that the supremum of $(a,b)$ is $b$.