Proof for absolute value inequality of three variables: $|x-z| \leq |x-y|+|y-z|$
5369
Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
77576
Using the triangle inequalities to prove a bound in the complex plane
49240
$\forall x,y \in \mathbb{R}:\vert\cos x- \cos y\vert\leq\vert x-y\vert$
15232
Proving an inequality involving real powers
7775
Show that $|x-y|\leq |x|+|y|$.
126632
Proving Inequalities with Absolute values
65297
Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2} \right |\leq \left |b-c \right |$.
126165
Prove $|\arctan x - \arctan y|\leq |x-y|$
137791
Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$
28218
Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$
1305
Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$
52204
Proof of one inequality $a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$
80789
Prove that $\forall a,b,c,d\in \mathbb{R}\,$, $(a^2-b^2)(c^2-d^2)\leq(ac-bd)^2$ and equalities hold if and only if $ad=bc$.
29935
If $a, b, c, d>0$ and $abcd=1$ prove that an inequality holds true
125209
Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
89170
How prove this inequality $\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y}\ge x^2+y^2+z^2$
15237
Prove an inequality using complex analysis