Proving that $\forall n\in \mathbb N, 2^n |(2n)!$
44906
Use induction to prove that $2^n$ divides $(2n)!$ for any $n\in\mathbb{Z}_{\geq0}.$
39175
Prove by induction that $\forall n \in\mathbb N$, $3\mid4^{n}-1$.
153763
Prove that $a_1!\cdots a_k! < n!$ whenever $a_1+\cdots+a_k < n$
124240
Inductive proof of $\,9 \mathrel| 4^n+6n-1\,$ for all $\,n\in\Bbb N$
68400
Proof that $\frac{(2n)!}{2^n}$ is integer
141088
Proof : $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$.
131771
Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$
150156
Proving that $S_n$ has order $n!$
20691
prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
136274
$n! \leq \left( \frac{n+1}{2} \right)^n$ via induction
45645
Using permutation or otherwise, prove that $\frac{(n^2)!}{(n!)^n}$ is an integer,where $n$ is a positive integer.
42292
Show that $a_n<\sqrt{2}$ for every $n\in\mathbb{N}$.
35611
How can I show that $n! \leqslant \left(\frac{n+1}{2}\right)^n$?
9920
$(n^2 - 1)!$ is a multiple of $(n!)^n$
145519
Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$
131754
Prove $ \forall c \in \mathbb{R^{+}}, \forall B \in \mathbb{N}, \exists n \in \mathbb{N}, (n \geq B) \wedge (\sqrt{2n} > c \ln(n+2)) $?
24403
Prove that $ n < 2^{n}$ for all natural numbers $n$.