Prove: if $x,y\in\mathbb{R}$ then $2xy\le x^2+y^2$
27911
How can I prove that $xy\leq x^2+y^2$?
61472
If $0<x<y$, then prove that $\sqrt{x} <\sqrt{y}$ and $x <\sqrt{xy} <y$
125707
A simple proof (hopefully) for $2^x \neq 0$
94119
Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$
58943
If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$
12484
Prove that $\lfloor 2x \rfloor + \lfloor 2y \rfloor \geq \lfloor x \rfloor + \lfloor y \rfloor + \lfloor x+y \rfloor$ for all real $x$ and $y$.
72313
Show that for all real numbers $a$ and $b$, $\,\, ab \le (1/2)(a^2+b^2)$
2315
How to prove that $\lVert x + y\rVert = \lVert x\rVert + \lVert y \rVert \implies \lVert tx + (1-t)y\rVert = t\lVert x\rVert + (1-t)\lVert y \rVert$?
144589
Proving that if $f(x) \le g(x)$ then $f(x_0) \le g(x_0)$
91877
Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
7775
Show that $|x-y|\leq |x|+|y|$.
99676
Prove $\sum{\frac{1}{(x+2y)^2}} \geq\frac{1}{xy+yz+zx}$
143072
Proving $|\sin x - \sin y| < |x - y|$
138931
Show that $||x+y||\cdot||x-y||\le||x||^2+||y||^2$
122944
Prove $|y''(x)| \leq 40$ for all $x \in [1,3]$.
31937
How do I prove equality of x and y?
105047
Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$