# Help with integral/logarithm inequality 1/(n+1)<int_n^(n+1)1/t

I have to prove the following inequality:
$1/\left(n+1\right)<{\int }_{n}^{n+1}1/t$$dt$$<1/n$
I thought it would be easier to attack this via integration, so I get:
$1/\left(n+1\right)<$ log $\left(n+1\right)-$ log$\left(n\right)<1/n$
At this point I tried to use induction, but the solution is still not clear to me.
Thanks a lot!
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Farbwolkenw
Hint: Draw a picture. On the interval $\left[n,n+1\right]$, our function $\frac{1}{t}$ is $\le \frac{1}{n}$, and $\ge \frac{1}{n+1}$
So the area under the curve $\frac{1}{t}$, and above the $t$-axis, from $t=n$ to $t=n+1$, is less than the area of a rectangle with base $1$ and height $\frac{1}{n}$, and greater than the area of a rectangle with base $1$ and height $\frac{1}{n+1}$
From the above geometric argument, it follows that $\frac{1}{n+1}<\mathrm{log}\left(n+1\right)-\mathrm{log}n<\frac{1}{n}$