How to prove/show that the sequence $a_n=\frac{1}{\sqrt{n^2+1}+n}$ is decreasing? How to prove/show that the sequence $a_n=\frac{1}{\sqrt{n^2+1}+n}$ is decreasing?
My idea:


*

*$n^2<(n+1)^2 /+1$

*$n^2+1<(n+1)^2+1/ \sqrt{}$

*$\sqrt{n^2+1}<\sqrt{(n+1)^2+1}/+n$

*$\sqrt{n^2+1}+n<\sqrt{(n+1)^2+1}+n$


And now I'm stuck since if I add 1 to the both sides, I don't know how to move it from the right side without also moving it from the left side.

Steps:
1) The sequence is decreasing if the denominators are increasing
2) $\sqrt{n^2+1}+n$ is increasing if both $\sqrt{n^2+1}$ and $n$ are increasing
3) $n$ is increasing. $\sqrt{n^2+1}>n$ is also increasing. Q.E.D.
