Are there sequences of positive real numbers converge to negative real number?

Question

I'm asking this because I'm having a hard time to prove the case of $x \lt 0$ for this question: Suppose that $x \in R, X_{n} \geq 0, and X_{n} \rightarrow x \: as \: n \rightarrow \: \infty$, Prove that $\sqrt{X_{n}} \: \rightarrow \: \sqrt{x} \: as \: n \: \rightarrow \: \infty$, since the assumption of $x \in R$ means x can be negative. But please don't directly answer my homework for me. I'm hoping to find if such assumption is correct first then continue my work for it. Thanks!

Answer 1

That cannot happen.

Suppose $x_n \to y$, where $y<0$ and $x_n >0$ for all $n$. Consider the interval $(y-\epsilon, y+\epsilon)$ around $y$, where $\epsilon < \frac{|y|}{2}$. In this neighborhood, can any $x_n$ be there?

Answer 2

If $x_n \to x$ and $x_n \ge 0$ for all $n$, then $x \ge 0.$

Suppose to the contrary that $x<0.$ Now take $ \epsilon >0$ with $x+ \epsilon <0$. Then there is $N$ such that

$$x_n <x+ \epsilon <0$$

for all $n>N.$ A contradiction !