Show that $\mathbb{E}~|\xi - \mathrm{med}(\xi)| \leq \mathbb{E}~|\xi - \mathbb{E}(\xi)| $
34797
$E|X-m|$ is minimised at $m$=median
15262
Prove $\exists \xi, \eta \in (0,1)$, such that $f(\xi)+f'(\eta)=\xi+\eta$.
113778
Show that $E=\{(x,\alpha)\mid 0\leq \alpha<|f(x)|\}$ is measurable if $f$ is measurable.
57096
Show that $\max{\{|a|+|b|,|c|+|d|\}} \leq \max{\{|a|,|c|\}}+\max{\{|b|,|d|\}}.$
73367
On $E[E[Y|X]|X]= E[Y|X]$
27347
Verify that $\left|\int_{\gamma} \exp(iz^2)dz\right| \leq \frac{\pi\big(1-\exp(-r^2)\big)}{4r}$ where $\gamma(t)=re^{it}$, for $0\leq t \leq \pi/4$.
108995
Prove $|e^{i\theta} -1| \leq |\theta|$
72506
An example s.t. $E(E(X|\mathcal F_1)|\mathcal F_2)\neq E(E(X|\mathcal F_2)|\mathcal F_1)$
20467
Show that $(y_n)\in l^q$
4656
Prove that there is a $\delta$ such that $\int_{0}^{1} (f(x))^2dx\leq \delta$$\int_{0}^{1} (f'(x))^2dx$ for all $f$ with these conditions
349
Prove $| \sum_{i \leq n} \frac{\mu(i)}{i} | \leq 1$
118613
Proving an inequality: $|1-e^{i\theta}|\le|\theta|$
36485
$H_n(X,A) \not \approx \tilde{H}_n(X/A)$
139711
prove this inequality with $\sum\limits_\text{cyc}\sqrt{1-xy}\ge 2$
105145
Show that $\mathbb{E}\left(\bar{X}_{n}\mid X_{(1)},X_{(n)}\right) = \frac{X_{(1)}+X_{(n)}}{2}$
99165
How to prove that for Brownian motion in $(a, b)$ $\mathbb{E}^x[\min(H_a, H_b)] = (x-a)(b-x)$?
51267
Showing a generalized version of the Hölder inequality for $L^{p}$ spaces