Simplest proof that $\binom{n}{k} \leq \left(\frac{en}{k}\right)^k$
2708
Inequality ${n \choose k} \leq \left(\frac{en}{ k}\right)^k$
37569
Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $
142589
Prove inequality of binomial theorem $\frac1{n^i}\binom{n}i \leq\frac1{m^i}\binom{m}i $.
25921
Show that $\left(1+\dfrac{1}{n}\right)^n$ is monotonically increasing
67927
Combinatorial argument for the identity $k\binom{n}{k} = n\binom{n-1}{k-1}$
98768
Prove $\left(1+\frac{x}{n}\right)^n < e^x$, where $x$ is any positive real number and $n$ is any positive integer.
40784
$\sum_{k=m}^{n}(-1)^k\binom{n}{k}\binom{k}{m}=0, n>m\geq 0$
9046
Prove that $\sum_{k=0}^{m}\frac{\binom{m}{k}}{\binom{n}{k}}=\frac{n+1}{n+1-m}$
63510
Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$
39073
Proving that: $\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$
10902
Showing that $ \prod\limits_{i=0}^{n} {n \choose i}\leq(\frac{2^n-2}{n-1})^{n-1} $
45742
Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$
77091
Inequality $\binom{2n}{n}\leq 4^n$
66247
Proving inequality $\left( \frac {x} {y}\right) ^{x}\left( \frac {1-x} {1-y}\right) ^{1-x}\geqslant 1$
155769
Prove $\sum_{k= 0}^{n} k \binom{n}{k} = n \cdot 2^{n-1}$ using the binomial theorem for $n\geq1$
21921
Showing that $|f(z)| \leq \prod \limits_{k=1}^n \left|\frac{z-z_k}{1-\overline{z_k}z} \right|$
59905
Proving the inequality $\frac{n^k}{k^k}\leq\binom{n}{k}\leq\frac{n^k}{k!}$ involving binomial coefficient.