How to prove that $|\cos x-\cos y|\leq |x-y| \ \forall x,y \in \mathbb{R}$?
49240
$\forall x,y \in \mathbb{R}:\vert\cos x- \cos y\vert\leq\vert x-y\vert$
129847
Show that $\forall (x,y)$ in the first quadrant: $\frac {x^2+y^2}{4}\leq e^{x+y-2}$
33543
is "y1 < x < y2" an inequality or inequalities?
139537
Show that $x\subseteq y \Leftrightarrow \mathcal{P}(x)\subseteq\mathcal{P}(y)$.
4498
Proof that $\|fx\| \leq \|f\|\cdot\|x\|$
50273
How to prove $\lvert \lVert x \rVert - \lVert y \rVert \rvert \overset{\heartsuit}{\leq} \lVert x-y \rVert$?
65297
Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2} \right |\leq \left |b-c \right |$.
27911
How can I prove that $xy\leq x^2+y^2$?
77026
Proof with inequalities
34223
Prove that every function that verifies $|f(x)-f(y)|\leq(x-y)^2$ for all $x,y$ is constant.
3820
Prove that $||x|-|y||\le |x-y|$
135895
Proving the bounds of cosine
126165
Prove $|\arctan x - \arctan y|\leq |x-y|$
8207
Can I prove this inequality algebraically?
76863
How to show $ \sup \inf g(x,y) \leq \inf \sup g(x,y)$?
108704
Help with proof of $|x+y| \geq |x|-|y|$
126632
Proving Inequalities with Absolute values