How about an intuitive approach which hopefully might lead to a more rigorous proof?
By the Mean value Theorem:
and surely as n goes to infinity, must necessarily migrate to the maximum value of . Suppose we could prove that part. Then if we take the n'th root of the expression, then in the limit and we're left with where M tended to the maximum value of f.
I know, it's lame and unfit to be in an Analysis forum: "Fight and you may die, run and you will live . . . for a little while."