Talk:Set-theoretic limit

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The first definition guarantees a limit, that limit being "at least" the empty set, {}. The second definition seems to suggest that it is possible for the limit superior to be different from the limit inferior, in which case the limit does not exist? Is it possible for the lim. inf. to be unequal to the lim. sup.? Doesn't this contradict the fact that the first definition guarantees the existence of a limit? Or is it that it is impossible for the two to be unequal? If so, that should be mentioned to avoid confusion, and perhaps the definitions should be presented as three different definitions: one using indicator functions, one using lim inf, and the other using lim sup. Or maybe the two definitions (one using indicator functions, and the other using lim inf and lim sup) are entirely different things, in which case this different should be highlighted, again, to avoid confusion.

EDIT: In fact, it has been shown to me that the first definition (using indicator functions) is equivalent to the given definition for "lim inf", and that indeed "lim sup" and "lim inf" can be different. So perhaps you should simply give one definition for the limit, that being the lim inf and the lim sup ONLY WHEN they are equal, and offer the characteristic function definition as an equivalent way to compute "lim inf." I would change it myself, but I'd rather have someone more knowledgeable actually do it (I don't actually know what the definition for a set-theoretic limit is, I just noticed the inconsistencies in the given ones).