Prove that if $0\le p_n \lt 1$ and $S:=\sum p_n \lt 1$, then $\Pi (1-p_n) \ge 1-S$.
73375
Show that $\prod (1- P(A_n))=0$ iff $\sum P(A_n) = \infty$
122115
Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$
153226
Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$
98662
$P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$
84559
Showing $\sum\limits_{p \in P} \frac{1}{p}$ where $P$ is the set of all primes is divergent
64108
how to prove $a+b-ab \le 1$ if $a,b \in [0,1]$?
66535
How to prove $P(A\cap B)\geqslant P(A)+P(B)-1$?
14197
show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
63879
Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
17157
Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
106767
$|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$
69921
Prove that: $(1+a_1)(1+a_2)...(1+a_n) \le 1 + S_n + (S_n)^2/2! + ... + (S_n)^n/n!$
3599
show that if ${a_n} \to a$, and $a>0$, then $\exists N$ such that $a_n>0$ for $n \ge N$.
43885
Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$
35529
Prove that $\sin x+2x\ge\frac{3x(x+1)}{\pi}$ for all $x\in [0,\frac{\pi}{2}]$
26741
How to prove for $s<1,|a+b|^s\le|a|^s+|b|^s$
158863
How to prove $\sum_p p^{-2} < \frac{1}{2}$?