Prove $\sup(A)\leq\inf(B)$
136789
How to prove $a \lt b$ for all $a \in A, b \in B$ implies $\sup A \leq \inf B$?
108995
Prove $|e^{i\theta} -1| \leq |\theta|$
132971
Prove that $|a| < b \iff -b<a<b$
5369
Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
89429
Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $
106767
$|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$
77855
Prove that if $A$ has a maximum, then the maximum must be the supremum
88687
Proof of the inequality $||a+b|^p - |a|^p| \leq \epsilon |a|^p + C_{\epsilon} |b|^p$
96931
Supremum and infimum of $\{\frac{1}{n}-\frac{1}{m}:m, n \in \mathbb{N}\}$
106979
Prove the inequality and limitation
96133
Prove that $ m_{a} \geq m_{g} \geq m_{h} $ using strict inequalities unless $ a = b $.
64320
Prove that if $0<a<Pr(A)$ then there exists $B\subset A$ such that $Pr(B)=a$
77192
How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
40524
Let $A \subseteq (0,\infty$) such that $\inf A>0$. Let $\frac{1}{A}=\{\frac{1}{a},a\in A\}$. Prove that $\sup \frac{1}{A}=\frac{1}{\inf A}$.
137511
Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$
26741
How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
27564
Prove that $\|UVU^{-1}V^{-1}-I\|\leq 2\|U-I\|\|V-I\|$