Convergence of sequence of absolute terms

Question

If the sequence $\{s_n\}$ converges to lets say $s$, then can we say that $\{|s_n|\}$ converges to $|s|$?

Answer 1

Yes, because $(\forall n\in\mathbb N):\bigl\lvert\lvert s\rvert-\lvert s_n\rvert\bigr\rvert\leqslant\lvert s-s_n\rvert$.

Answer 2

Yes. Because $y=\mid x\mid$ is continuous.

Answer 3

Yes since $||x|-|y|| \leq |x-y|$ for all $x,y \in \Bbb{R}$

But we cannot say that $|x_n| \to |x| \Longrightarrow x_n \to x$

Take for instance $x_n=(-1)^n$