How to prove $(\frac{n+1}{n})^n

Question

Prove that $(\frac{n+1}{n})^n<n\quad $ for any n=3,4,5... by using induction.

For n=3 is true. and lets assume $(\frac{n+1}{n})^n<n$ is true. we must show that $(\frac{n+2}{n+1})^{n+1}<n+1$ is true. How can I continue?

Answer 1

If $(\frac{n+1}{n})^n<n$ is true, then

$(\frac{n+1}{n})^n*\frac{n+1}{n}<n*\frac{n+1}{n}$ is also true which is $(\frac{n+1}{n})^{n+1}<n+1$

since $\frac{n+1}{n}>\frac{n+2}{n+1}$

$(\frac{n+2}{n+1})^{n+1}<n+1$ is true.