n^(n+1)>(n+1)^n的數學歸納法(急)

設 n=k時,k^(k+1) > (k+1)^k………..(1)
(由(k+1)^2>k(k+2)…..得(k+1)/k > (k+2)/(k+1)
得[(k+1)/k]^(k+1)> [(k+2)/(k+1)] ^(k+1)…..(2)
(1)*(2) 得(k+1)^(k+2) > (k+2)^(k+1)

I think it is impossible to use M.I. to prove the required result. Here is my proof :

n^(n+1) > (n+1)^n <=> n > (1+1/n)^n
As we know that a_n=(1+1/n)^n is an increasing sequence which is bounded by e < 3. So n^(n+1) > (n+1)^n when n >=3