Is zero absolute limit equivalent to zero limit?

Question

Is is true that:

$\lim$ $|a_n| = 0$ $\equiv \lim$ $a_n = 0$

I know that

$\lim$ $|a_n| = 0$ $\leftarrow \lim$ $a_n = 0$.

But what about the other direction?

Thanks very much

Answer 1

Yes use the squeeze (or sandwich) theorem

Suppose $\lim a_n=0$ then $\lim -a_n=0$ by the constant multiple rule and for each $n$ we have one of the following two inequalities

$$ -a_n \le |a_n| \le a_n $$

or

$$ a_n \le |a_n| \le -a_n $$

so by the squeeze theorem $\lim a_n=0$

Answer 2

Suppose that $|a_n|\to 0$. Then given $\epsilon>0$, there exists an integer $N$ such that $n> N$ implies that $$ |a_n-0|=||a_n|-0|<\epsilon $$ as desired.

Answer 3

Recall that

$$-|a_n|\le a_n \le |a_n|$$

then by squeeze theorem we can conclude that

$$\lim_{n\to \infty}|a_n| = 0 \implies \lim_{n\to \infty} a_n = 0$$

Answer 4

Yes on the question in the title.

The following statements are equivalent because $||a_n|-0|=||a_n||=|a_n|=|a_n-0|$

• $\forall\epsilon>0\exists n\in\mathbb N\forall k>n \left[||a_n|-0|\leq\epsilon\right]$

• $\forall\epsilon>0\exists n\in\mathbb N\forall k>n\left[|a_n-0|\leq\epsilon\right]$