# Why do we use the absolute value when defining a limit?

I am having a hard time understanding the definition of a limit.

Specifically, why we define convergence to L (when there exists N such that for all $$n>N$$) as

$$|s_n-L| <\epsilon$$

$$s_n-L<\epsilon$$

I have been thinking about this for a while and I can't seem to wrap my head around why we need the absolute value.

• What if $s_n - L$ is $-1000$? Then $s_n - L < \epsilon$ yet $s_n$ is nowhere near $L$. – littleO Sep 21 '18 at 0:51

We want a statement that says, when $$n$$ is large enough, then the distance between $$s_n$$ and $$L$$ get smaller and smaller.
Think of the case $$s_n = \displaystyle{\sum_{k = 1}^{n}} -\frac{1}{2^k}$$. It clearly approaches $$-1$$ and the distance $$|s_n - (-1)|$$ gets arbitrarily close to $$0$$, but if we used $$s_n - (-1)$$, it doesn't make sense to say "distance" since it will be negative. The absolute value is just a convenience, so we don't have to worry about negatives and can say that the distance gets arbitrarily close to $$0$$.