so i started studying calculus and stumbled upon an inequality I'm having difficulties to prove,

for every 1<=k<=n prove

C(n,k)*1/(n^k)<=1/(2^(k-1))

so i brought it to

C(n,k)*2^k<=2*n^k

and now i tried induction and it got to newton's binomial, and then im stuck...

looks something like

C(n,k-1)*2^k+c(n,k)*2^k<=2*(1+n+n^2...kn^k-1+n^K) used c(n+1,k)=c(n,k-1)+c(n,k)

(dont know how to write epsilon so its open ^^^^^ )

while the red thingy is the assumption of the induction,

how i prove the pink part??

please, require assistance!