Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$
6100
Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$
90343
Proving the inequality $|e^z-1|\leq e^{|z|}-1$
86535
Prove quotient of two $N(0,1)$ is $\text{Cauchy}(0,1)$
45219
Proving that $C$ is a subset of $f^{-1}[f(C)]$
95513
Proving $W$ is a subspace of $P_{2}$
158524
Complex numbers $z$ such that $|z|= 1$
17508
Knowing $|z|$ and $|w|$ (and possibly $z\overline{w}$), what are $|z+w|^2$, $|zw|^2$, $|z-w|^2$, $|z/w|^2$?
132957
Prove that the derivative of $x^w$ is $w x^{w-1}$ for real $w$
83713
If $z,w\in \mathbb{C}^{\times}$, then $|z+w|=|z|+|w|$ iff $w=tz$ for some $t\gt 0$
46039
Prove that if $x \in R,$ then there exists $n \in Z$ satisfying $x \leq n < x+1$
108995
Prove $|e^{i\theta} -1| \leq |\theta|$
64096
Prove that $||z_1|-|z_2|| \leq |z_1-z_2|$.
34308
Given $|z|<1$ prove $\lim_{n \to \infty} z^n=0$.
74202
Show that $|f(z_1)|=|f(z_2)|≤|z_1|^2=|z_2|^2$.
91877
Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
43246
Showing that if $a<b$, then $a^{1/n} < b^{1/n}$
84946
If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$