Prove that $ \frac{1}{1+n^2} < \ln(1+ \frac{1}{n} ) < \frac {1}{\sqrt{n} }$ For $n >0$ ,
Prove that $$ \frac{1}{1+n^2} < \ln(1+ \frac{1}{n} ) < \frac {1}{\sqrt{n}}$$
I really have no clue. I tried by working on  $ n^2 + 1 > n  > \sqrt{n} $ but it gives nothing.
Any idea?

This inequality is correct only for $n >1$ and not for $n >0$ as asked by the person.
For $n >1$, we know that $$\tag1 \frac{1}{n}<\frac{1}{\sqrt n}$$ and $$\tag2 \frac{1}{(1+n^2)} < \frac{1}{(1+n)}$$
Also, $\tag3\frac1{n+1}<\ln\left(1+\tfrac1n\right)<\frac1n, \forall n>0$
So the overall inequality $$\tag4\frac{1}{(1+n^2)} < \frac{1}{(1+n)}<\ln{\left(1+\frac1n\right)}<\frac{1}{n}<\frac{1}{\sqrt n}$$ will be valid only for $n>1$ and not for $n>0$.
