Basics of Limits: Why Absolute Value?

[Born2Perform]

I know its a banal question but I am new to calculus course and there are no math teachers in summer... also my book does not explain it

Why it is
$$\mid a_n - L \mid < \varepsilon$$

And not just
$$L - a_n < \varepsilon$$

??
wouldn't the relation be the same?

 

[quasar987]

If the definition of convergence did not include the absolute values, it would mean, for instance, that the constant sequence $a_n = L+1$ converges to L since for all n, $L-(L+1)=-1<\epsilon$.

 

[benorin]

The statement $$\mid a_n - L \mid < \varepsilon$$ requires $$a_n$$ to be no more than $$\varepsilon$$ away from L, that is to say $$a_n\in (L- \varepsilon , L+ \varepsilon )$$;
whereas the statement $$L - a_n < \varepsilon$$ only requires $$a_n$$ less than L by no more than $$\varepsilon$$, that is to say $$a_n\in (L- \varepsilon , L)$$.

 

[Office_Shredder]

Here's a more intuitive way of looking at it.. limits are all about showing two numbers get really close together...

The best (only) way to measure distance in one dimension is absolute value. So of course you'd use that

 

[HallsofIvy]

As office shredder said, |a- b| is the distance between numbers a and b. Of course, a distance is never negative and the distance from a to b is the same as the distance from b to a so it shouldn't matter which is larger. The number 7 is 3 steps away from the number 4. I could see that, of course by subtracting 7- 4= 3 but when I am using variables, I don't know which is larger. |7- 4|= |4- 7|= 3 regardless of which is larger.