If {$b_n$} converges to b, then prove that {|$b_n$|} converges to |b|.

Question

Is my proof good or does it need more work? Let $\epsilon$ > 0, we want N s.t. $\forall$ n $\geq$ N $\subset$ ||$b_n$| - |b|| < $\epsilon$. If |$b_n$| $\longrightarrow$ |b|, then $\exists$ N $\in$ $\mathbb{N}$ s.t. |$b_n$ - b| < $\epsilon$ $\forall$ n $\geq$ N. $\therefore$ {$b_n$} $\longrightarrow$ b and {|$b_n$|} $\longrightarrow$ |b|

Answer 1

$ ||x| - |y||\leq|x - y| $ thus for that N we have $\forall$ n $\geq$ N
|$|b_n| - |b||\leq |b_n - b| < \epsilon$

Answer 2

$|a-b| \ge ||a|-|b||$ for any $a$ and $b$.

Answer 3

An other way

$x\mapsto |x|$ is continuous at $\ell$, and thus $$\lim_{n\to\infty }|b_n|=|\lim_{n\to\infty }b_n|=|\ell|.$$