If $1\leq p < r \leq \infty$ , then $\| x \|_p \geq \| x \|_r $
20844
Relations between p norms
13561
Show: $E(|X|)\leqslant\liminf_{n\to\infty} E(|X_n|)$
95293
Proof that $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$
73450
Show that $\sum_{n=1}^{\infty}X_n<\infty$ almost surely if and only if $\sum_{n=1}^{\infty}\mathbb E[\frac{X_n}{1+X_n}]<\infty$.
116966
Prove $\limsup_{n \rightarrow \infty} a_n\geq1$ if $\lim_{n \rightarrow \infty} a_na_{n+1}=1$
32063
if $f'(x)\rightarrow L$ as $ x \rightarrow \infty$, $-\infty \leq L \leq \infty $ then $ f(x)/x \rightarrow L $ as $x \rightarrow \infty$
42150
Prove that $\forall x \neg P(x) \iff \neg \exists x P(x)$
76689
How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?
82591
Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.
48593
Proof that if $s_n \leq t_n$ for $n \geq N$, then $\liminf_{n \rightarrow \infty} s_n \leq \liminf_{n \rightarrow \infty} t_n$
109778
Proving $(x-c)|(p(x)-p(c))$
55019
Lebesgue density theorem in the line
67851
Does $|x|^p$ with $0<p<1$ satisfy the triangle inequality on $\mathbb{R}$?
28125
Showing that $1 - \frac{x^2}2\leq\cos x$, $\forall x \in \mathbb{R}$
102079
If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic
46039
Prove that if $x \in R,$ then there exists $n \in Z$ satisfying $x \leq n < x+1$
106767
$|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$