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?

I found that with 
$$ ln(1+x) < x $$  $\forall x\in ]0,+\infty[$
we can easily prove the RLH side of inequality, because $sqrt(x) < x$
still have the problem of the first one.
oh I found something else :
we have also 
$$ \frac{x}{(1+x)} < ln(1+x) $$  $\forall x\in ]0,+\infty[$
thus here we have $$ ln(1+1/n) >\frac{1}{(1+n)} > \frac{1}{(1+n^2)} $$
I think I have answered my question. thank you :) 
