If sequence $x_n$ converges to $x$, prove that $\sqrt x_n$ converges to $\sqrt x$

Question

If sequence $x_n$ converges to $x$, prove that $\sqrt x_n$ converges to c. I know I have to estimate

$| \sqrt x_n - \sqrt x |$. But I cannot start. Thanks

Answer 1

Hint :
If your $x$ is zero, it is trivial.
Else, note that $$|\sqrt a-\sqrt b|=\frac{|a-b|}{\sqrt a+\sqrt b}$$ So does $(\sqrt x_n+\sqrt x)$ have a lower limit here?

Answer 2

If $f$ is a continuous function and $\{a_n\}$ a sequence that converges to a limit $c$ , it is a useful fact that $\{f(a_n) \}$ converges to $f(c)$.

You should prove this, as it comes up a lot. Besides, it'll be easier to prove the more general case, and then apply it to this specific problem.