Why does the absolute value of a power series converge to the same limit as the power series?

Question

Power series, if $$\lim_{{k}\to\infty}ka_kx^{k-1}=0$$ for any $x\in(-1,1)$

Why does this imply $$\lim_{{k}\to\infty}|ka_kx^{k-1}|=0$$ for any $x\in(-1,1)$

I would normally show what work I've done to try and solve for myself but in this case I just don't know what to say. It seems obvious that if the limit goes to zero then the absolute value would also go to zero but I don't know how to formally state this. Is there something deeper that I'm not seeing, why would this be useful/important?

Any help would be greatly appreciated.

Thank you,

Answer 1

$$x=0\implies \vert x\vert=0$$ By definition of $\vert x\vert$.

Answer 2

The map $u\rightarrow|u|$ is continuous. Perhaps, one sees that $\big||\cdot|-|u|\big|\leq|\cdot-u|$.