Can we remove the absolute value from inequality $|a-b|<ε$?

Question

So currently I have $|S_n-S|<ε$ where $S_n$ is a sequence and $ε>0$.

I have $\bigl||S_n|-|S|\bigr|\leq|S_n-S|<ε$. Hence, $\bigl||S_n|-|S|\bigr|<ε$. Since $ε>0$, can I just remove the main absolute value and say $|S_n|-|S|<ε$?

Answer 1

Yes, you always can do that because $|x|< a \iff x<a$ and $x>-a$, so in particular you can remove absolute value.

Also observe that is always true that $x<|x|$, hence if $|x|<a$ we conclude that $x<a$

Answer 2

Just to spell out Robson's argument:

If $|S_n| -|S|$ is positive then it's equal to $\bigg||S_n| -|S| \bigg| <\varepsilon$ and if $|S_n| -|S|$ is negative then it's certainly less than $\epsilon$ which is positive.