Prove that the inequality `|sinn x|len|sinx|` is valid for all positive integers `n`

Prove that 2^n > n for all positive integers n.

Prove that 2^n gt n for all positive integers n.

Prove that 2^(n) gt n for all positive integers n.

Prove that 2^(n) gt n for all positive integers n.

Prove that 2^(n) gt n for all positive integers n.

Prove that quad 2^(n)>n for all positive integers n.

Prove by induction the inequality (1+x)^ngeq 1+n x whenever x is positive and n is a positive integer.

If A=((cosx,sinx),(-sinx,cosx)) then prove that A^n=((cosnx,sin nx),(-sin nx,cosnx) for all positive integers n.

Prove that x^n-y^n is divisible by x - y for all positive integers n.

Prove that (n !)^2 < n^n n! < (2n)! , for all positive integers n.