TITLE: Why is the following sequence monotonically decreasing?
QUESTION [12 upvotes]: The sequence in question is 
$$
a_n = \sqrt{n^2+4}-n\,.
$$
I can see that it is strictly decreasing by finding the derivative and observing that it is negative on the entire range containing the relevant values of $n$. But I feel like there must be a simpler algebraic reason that I'm just not seeing.

REPLY [9 votes]: Write it as $a_n = \sqrt{n^2 + 4} - \sqrt{n^2}$.
Interpret it as "the increase in $\sqrt{x}$ when $x$ is increased by $4$", for $x = n^2$. Note that this is decreasing in $x$ iff it's decreasing in $n$, since all these numbers are positive.
Another way of thinking about this is as "the amount by which I need to increase $y = \sqrt{x}$ to make $y^2$ increase by $4$".
Since $y^2$ grows faster for large $y$, this amount is going to get smaller for large $y$.