Question

Prove that lim(Xn) =0 iff lim(|Xn|)=0 . Give an example to show that convergence of (|Xn|) need not imply the convergence of (Xn)

Answer 1

$\lim\limits_{n\to\infty}x_n=0\Leftrightarrow\forall\varepsilon>0\ \exists N\ \forall n>N\ |x_n-0|<\varepsilon$

$\lim\limits_{n\to\infty}|x_n|=0\Leftrightarrow\forall\varepsilon>0\ \exists N\ \forall n>N\ ||x_n|-0|<\varepsilon$

Since $|x_n-0|=||x_n|-0|$, we obtain that $\lim\limits_{n\to\infty}x_n=0\Leftrightarrow\lim\limits_{n\to\infty}|x_n|=0$

Example of a divergent sequence $x_n$ such that $|x_n|$ is a convergent sequence: $x_n=(-1)^n$