Prove inequality by MI

Prove n^(n+1)>n(n+1)^(n-1) for n>1, where n is an integer.
Let S(n) be the statement.
"n^(n+1)>n(n+1)^(n-1)"
when n=2,
L.H.S=2^3=8
R.H.S=2(3)^1=6
L.H.S>R.H.S
S(1) is true.
Assume S(k) is true.
when n=k
k^(k+1)>k(k+1)^(k-1)
k(k/k+1)^(k-1)>1
when n =k+1,
since
k(k/k+1)^(k-1)>1 [by assumption]
(k+1)(k/k+1)^k=k(k/k+1)^(k-1)>1
(k+1)(k+1/k+2)^k>(k+1)(k/k+1)^k>1
(k+1)^(k+1)>(k+2)^k
(k+1)^(k+2)>(k+2)^(k+1)
S(k+1) is also true.
By the principle of mathematical induction,S(n) is true for all positive n except 1.

For n <2
S(n) is not true.

P(n) : n^(n + 1) > n(n + 1)^(n - 1)
for all integers n.

When n = 1 :
L.S. = 1^2 = 1
R.S. = 1 x 2^0 = 1
Hence, L.S. = R.S. and L.S. not > R.S.

P(1) is false when n = 1.
Hence, P(n) is false.