Would it be correct to define the limit of a series as the smallest number that no number in the series is greater than (for an increasing series, the other way around for a decreasing series, i.e the greatest number that no number in the series is smaller than)?
EDIT: Based on Hagen's comment I realized I meant sequence, not series; I wasn't clear on the difference.
Yes, but such a definition would not extend easily to sequences which are not monotone. Some examples are $(-1)^n$ which does not have a limit, but would have limit either 1 or -1 based on your definition, or $e^{-n}\sin(n)$, which has a limit of 0 but has no such limit based on either of your definitions.