Prove the following inequality for each natural $n$: $$(n+1)^{n-1} \leq n^n$$ In fact I am not sure if it is true, but at least for $n\leq 4$ it holds true. I tried induction, but with no success. Thanks in advance for help.
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4We have $\left(1+\frac{1}{n}\right)^n\lt 3$. So lots of slack! – André Nicolas Jan 08 '14 at 20:53
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See an inductive proof for a similar problem. – Anant Mar 22 '14 at 21:32
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Hint: For $n > 0$, the inequality is equivalent to
$$\left(1 + \frac1n\right)^{n-1} \leqslant n.$$
Daniel Fischer
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