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

Question

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

$|s_n-L| <\epsilon$

instead of

$s_n-L<\epsilon$

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.

Answer 1

Absolute value measures distance.

We want a statement that says, when $n$ is large enough, then the distance between $s_n$ and $L$ get smaller and smaller.

Answer 2

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 $0$.