Proving $x+y =\sup\left\{ r+s: r,s \in \mathbb{Q}, r\leq x, s\leq y\right\}$
14162
How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$
20626
Prove that $x^y < y^x$
125209
Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
5369
Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
94119
Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$
52095
Prove either $G=ST$ or |$G|\geq|S|+|T|$
1305
Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$
12083
How to prove $\limsup (x_{n}+y_{n})=\lim x_{n}+\limsup y_{n}$?
53660
Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q  \implies  \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $
63879
Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
98921
Does $y(y+1) \leq (x+1)^2$ imply $y(y-1) \leq x^2$?
28218
Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$
20467
Show that $(y_n)\in l^q$
152311
Proving expression$\geq 2$, where variables are $x,y,z$
66247
Proving inequality $\left( \frac {x} {y}\right) ^{x}\left( \frac {1-x} {1-y}\right) ^{1-x}\geqslant 1$
59872
How to prove $\inf(S)=-\sup(-S)$?
3820
Prove that $||x|-|y||\le |x-y|$