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

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 $$|S_n|-|S|<ε$$?

• 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/… – Robson Nov 24 '18 at 4:01
• Thanks for the advice! – Sashin Chetty Nov 24 '18 at 7:41

## 2 Answers

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

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

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.