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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 $|S_n|-|S|<ε$?

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    $\begingroup$ Hello @Sashin Chetty, for better questions and answer, here you have to use Math Jax to write math equations... here is a resume math.meta.stackexchange.com/questions/5020/… $\endgroup$ – Robson Nov 24 '18 at 4:01
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    $\begingroup$ Thanks for the advice! $\endgroup$ – Sashin Chetty Nov 24 '18 at 7:41
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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 $x<a$

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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.

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