A simple question about nonnegativity of square roots

Question

Say I have a sequence $(x_n)$ and also that $x_n \geq 0 $. Let $x = \lim_{n\to\infty}x_n$. I also know that $x \geq 0$. My question is,

1. Is $\sqrt{x_n} + \sqrt{x}$ a nonnegative quantity? I believe this is so because the square root operator returns the principal (nonnegative root)? Is this reasoning correct?

Answer 1

If $a,b \ge 0$, then $\sqrt{a},\sqrt{b} \ge 0$ hence $\sqrt{a}+\sqrt{b} \ge 0.$

Answer 2

Yes your reasoning is correct. The square root operator returns the principal (non-negative) root.