can you please elaborate and explain the answer forprove n^n<=(2n)! using mathematical induction?

given : n^n

to prove : n^n = 2n!

for n=1

(1^1 = 1) <= (2*1 = 1)

since it is true for n=1, for n=k

k^k <= 2*k!

assuming n=k to be true, multiplying bothsides by k we get

k^k * k^1 <= (2*k)! * k

k^k+1 <= (2*k*k) * (2*(k-1))! = 2[(k*k)*(k-1)]!

this is of the form  n^n = 2n!

therefore n^n = 2n!