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


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.

  • 4
    $\begingroup$ What if $s_n - L$ is $-1000$? Then $ s_n - L < \epsilon$ yet $s_n$ is nowhere near $L$. $\endgroup$ – littleO Sep 21 '18 at 0:51

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.


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 $0$.


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