How to find if a curve is bounded by two others? [duplicate]
Your inequality is equivalent to$$\frac1{n-1}>\log\left(\frac n{n-1}\right)>\frac1n,$$which, in turn, is equivalent to$$\frac1{n-1}>\log(n)-\log(n-1)>\frac1n.$$Now, use the fact that$$\log(n)-\log(n-1)=\int_{n-1}^n\frac{\mathrm dt}t.$$