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,

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?

You are taking the absolute value. Even if you don't take the absolute value, then also it is correct, as for any $a\ge 0\implies \sqrt{a}\ge 0$. — Sachchidanand Prasad, Feb 18 '21 at 12:43
Yes your reasoning is correct. The square root operator returns the principal (non-negative) root. — Adam Rubinson, Feb 18 '21 at 12:43
@SachchidanandPrasad just edited realizing that! Thank you for the reply. — anAnonymousEconomist, Feb 18 '21 at 12:46

score 2 · Answer 1 · answered Feb 18 '21 at 12:43

2

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

answered Feb 18 '21 at 12:43

Fred

score 1 · Answer 2 · answered Feb 18 '21 at 12:43

1

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

answered Feb 18 '21 at 12:43

Adam Rubinson

2 Answers2