Prove the estimate $\binom{n}{k} \le (\frac{en}{k})^k$
2708
Inequality ${n \choose k} \leq \left(\frac{en}{ k}\right)^k$
2885
Prove that $\sum\limits_{i=1}^n 2i\binom{2n}{n-i}= n\binom{2n}{n}$
107058
Prove that $E(X\ln X)\le E X E\ln X$
141446
Proving by induction that $\sum\limits_{k=2}^n \frac1{k^2}\le 1$
104807
Prove the inequality $\left|\frac{m}{n}-\frac{1+\sqrt{5}}{2}\right|<\frac{1}{mn}$
136274
$n! \leq \left( \frac{n+1}{2} \right)^n$ via induction
52557
Derivation of binomial coefficient in binomial theorem.
59638
How to prove this result on binomial coefficients?
40420
Proof by induction of Bernoulli's inequality: $(1 + x)^n \geq 1 + nx$
19804
How to prove that $|w_{n+1}-w_n| \leq (2/3)^2 |w_n - w_{n-1}|$?
94119
Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$
5049
Proving $\sum_{k=0}^n(-1)^k\binom nk=0$
35956
Proof that a sum of Bernoulli rvs has Binomial distribution
139674
Prove $\ln\frac{p}{q}\leq \frac{p-q}{\sqrt{pq}}$ for $0<q\leq p$
47827
Prove that $\sum^{n}_{r=0}(-1)^r\cdot \large\frac{\binom{n}{r}}{\binom{r+3}{r}} = \frac{3!}{2(n+3)}$
118559
Prove the inequality $\sum_{1\le i<j\le n}\sqrt{a_ia_j}<\frac{n-1}2\sum_{i=1}^n a_i$
16392
Prove the Inequality on sequence