Prove $|z_1|-|z_2|\leq|z_1-z_2|$
64096
Prove that $||z_1|-|z_2|| \leq |z_1-z_2|$.
13794
$|z_{1} + z_{2}| = |z_{1} - z_{2}| \implies z_{1}/z_{2}$ is Imaginary
59479
For a positive integer $n\geq 2$ with divisors $1=d_1<d_2<\cdots<d_k=n$, prove that $d_1d_2+d_2d_3+\cdots+d_{k-1}d_k<n^2$
155232
Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
51659
If $f(0)=0$ then prove that $|f(z)+f(-z)| \leq 2|z|^2$
81618
Prove $||a|-|b| | \leq |a-b|$
15237
Prove an inequality using complex analysis
41383
Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$
52800
Show that $z^5-1=(z-1)(z^2+2z\cos{\pi \over 5}+1)(z^2-2z\cos{2\pi \over 5}+1)$
146180
Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$
114867
Prove $\forall n\geq 2,n\in\mathbb{Z}$, $(n+1)\mid(n^3+1)$
149700
Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$
27564
Prove that $\|UVU^{-1}V^{-1}-I\|\leq 2\|U-I\|\|V-I\|$
139503
How to prove $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?
49054
Show that $\left | \sin a - \sin b \right | \leq \left | a-b \right |$
80009
Prove that $| |a|-|b| | \le |a-b|$
105641
Prove that $ab\leq\frac{1}{2\varepsilon}a^2+\frac{\varepsilon}{2}b^2$