# Determine if a sequence converges before finding the limit

I've got an exam in a couple of days and one of the exercises in my book is:

Determine the convergence of this sequence, then, if the sequence converges, find the limit. $$\sqrt{n-1} - \sqrt n$$

Now I know how I'd find the limit, that's not a problem. But how do I find out if the sequence converges before finding the limit, as the exercise implies I should do?

Edit: what I would probably do is find the limit, then say that the existance of a limit L in R implies that it's convergent

• Try multiplying above and below by $\sqrt{n-1}+\sqrt{n}$? Commented Apr 12, 2021 at 20:54
• Hint: What is $(\sqrt{n+1}-\sqrt{n})(\sqrt{n+1}+\sqrt{n})$? Commented Apr 12, 2021 at 20:56
• Yep, I would get 1 over something with n. Would that be a rigorous enough proof of convergence though? Thank you. Commented Apr 12, 2021 at 20:59
• I think you may be overhinking this: if you know how to find the limit, then your reasoning must show that the sequence converges. Perhaps you should show us your working for finding the limit. Commented Apr 12, 2021 at 22:41
• I figured that that's the case. You guys helped me in just the way I needed. Thank you Commented Apr 14, 2021 at 2:16

For example, you can show that your sequence is increasing and bounded above. Do you see how you can do this to $$\sqrt{n-1}-\sqrt{n}$$ ?
• $$\left\{\frac{(-1)^n}n\right\}\;$$ is not monotonic yet it converges to zero. And any convergent sequence is bounded. Commented Apr 12, 2021 at 20:56