a simple inequality for a monotone function

Question

To prove that the sequence $a_n = \sqrt{n^2+1} - n$ is monotone I used $a_{n+1}-a_n$ and got $\sqrt{(n+1)^2+1}-\sqrt{n^2+1}-1$, which is always negative, but that's as far as I'm able to process it, how do I show formally that it's less than $0$?

($n$ is in $\mathbb N_+$)

Answer 1

You have $a_n=1/b_n$ with $b_n=\sqrt{n^2+1}+n$, and $b_n$ is clearly $>0$ and increasing, so $a_n$ is decreasing.