Answer
Proof given below.
Work Step by Step
Let statement A: $\{a_{n}\}$ converges to $0.$
Let statement B: $\{|a_{n}|\}$ converges to $0.$
Prove A$\Rightarrow$B
If $a_{n}$ converges to $0$,
then for any $\epsilon\gt 0$,
there exists an index N such that $(n\gt N\Rightarrow|a_{n}-0||\lt \epsilon)$
$\Rightarrow|a_{n}|\lt \epsilon$
Since $||a_{n}||=|a_{n}|$, it follows that
$||a_{n}||\lt \epsilon$
$\Rightarrow ||a_{n}| -0|\lt \epsilon$.
meaning that $|a_{n}|$ converges to 0.
Prove B$\Rightarrow$A.
If $|a_{n}|$ converges to 0.
then for any $\epsilon\gt 0$,
there exists an index N such that $(n\gt N\Rightarrow||a_{n}|-0||\lt \epsilon)$
$\Rightarrow ||a_{n}||\lt \epsilon$
Since $||a_{n}||=|a_{n}|$, it follows that
$|a_{n}|\lt \epsilon$
$\Rightarrow|a_{n}-0|\lt \epsilon$,
meaning that $a_{n} \rightarrow 0$.
Since A$\Rightarrow$B and B$\Rightarrow$A, it follows that statements A and B are equivalent. A is true if and only if B is true.
This proves the problem statement.
