How do I prove that $n^n > (n+1)^{n-1}$? It seems simple enough, and by using brute force it's easy to see that $n^n$ will always be slightly larger for any $n \ge 3$. I tried comparing ratios and also using induction, but nothing is conclusive. I would imagine that the epsilon proof would be the correct way to do this but I'm not very familiar with it and I would rather avoid such arduous proofs when I have to solve similar problems in a limited time frame.

HINT: it is equivalent to $$n+1>\left(1+\frac{1}{n}\right)^n$$
