Absolute value and limits

Question

Could you give me, please, some counterexamples to the following statement:

$$ \lim_{n \to \infty} |X_n|= |a| \Rightarrow \lim_{n \to \infty} X_n = a $$

I know, that $$ \lim_{n \to \infty} X_n = a \Rightarrow \lim_{n \to \infty}|X_n|= |a|$$ because $|.|$ is a continuous function. However, despite I know, that the converse should be false, I can not construct any explicit counterexamples. Please, help me.

Answer 1

Let $(x_n) = (-1)^n$. Then $\lim_{n\rightarrow\infty} |x_n| = 1$, but $\lim_{n\rightarrow\infty} x_n $ does not exist.

Answer 2

A trivial counterexample follows:

$X_n = -a~\forall n\in\mathbb{N}$, then clearly $\lim_{n\rightarrow\infty}|X_n|=|-a|=|a|$ but $\lim_{n\rightarrow\infty}X_n=-a\neq|a|$

A nontrival counterexample follows:

$X_n = (-1)^n~\forall n\in\mathbb{N}$, then $|X_n|=1~\forall n\in\mathbb{N}$, so $\lim_{n\rightarrow\infty}|X_n| = \lim_{n\rightarrow\infty}1 = 1$. But $\lim_{n\rightarrow\infty}X_n$ does not exist (it merely "oscillates" between 1 and -1).