analysis - If in a normed space $a_n$ converges to $a$, prove that $||a_n||$ converges to $||a||$

If in a normed space an converges to $a$, prove that $||a_n||$ converges to $||a||$

I have seen this proof: for all $\varepsilon>0$ there exists and $N$ s.t. $n>N$ implies
that the $||a_n-a||<\varepsilon$

by the reverse triangle inequality for the same $N \varepsilon > ||an-a|| > |\ ||an||-||a||\ |$

and therefore it converges.

But this is the absolute value of the difference of norms. Don't you need the norm of the difference of norms to be less than $\varepsilon$?

As in

$||\ ||an|| - ||a||\ || < \varepsilon$

not

$|\ ||an|| - ||a||\ | < \varepsilon$

Or am I confused

Thanks!