$\infty - \infty = 0$ ? I am given this sequence with square root. $a_n:=\sqrt{n+1000}-\sqrt{n}$. I have read that sequence converges to $0$, if $n \rightarrow \infty$. Then I said, well, it may be because $\sqrt{n}$ goes to $\infty$, and then $\infty - \infty = 0$. Am I right? If I am right, why am I right? I mean, how can something like $\infty - \infty = 0$ happen, since the first $\infty$ which comes from $\sqrt{n+1000}$ is definitely bigger than $\sqrt{n}.$?

Note that 
$$\sqrt{n}\lt \sqrt{n+1000}\lt \sqrt{n}+\frac{500}{\sqrt{n}}.$$
This is because 
$$\left(\sqrt{n}+\frac{500}{\sqrt{n}}\right)^2=n+1000+\frac{500^2}{n}\gt n+1000.$$
If follows that
$$0\lt \sqrt{n+1000}-\sqrt{n}\lt \frac{500}{\sqrt{n}}.$$
Now by Squeezing we can conclude that
$$\lim_{n\to\infty}(\sqrt{n+1000}-\sqrt{n})=0.$$
