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Does the lim sup always exist?
lim sup and lim inf always exist (possibly infinite) for any sequence of real numbers. It is important to try to develop a more intuitive understanding about lim sup and lim inf. The next results will attempt to make these concepts somewhat more clear.
Is lim sup lim?
Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.
Can limit superior be infinity?
Yes. Sort of the whole point to lim sup and lim inf is that they are defined for any sequence of reals. You need to keep in mind that ∞ and −∞ are allowed as values.
What is a sup function?
The supremum (abbreviated sup; plural suprema) of a subset of a partially ordered set is the least element in that is greater than or equal to all elements of if such an element exists. Consequently, the supremum is also referred to as the least upper bound (or LUB).
What does sup mean in mathematics?
Sup (“supremum”) means, basically, the largest. So this: supk≥0T(k)(N) refers to the largest value T(k)(N) could get to as k varies. It’s technically a bit different than the maximum—it’s the smallest number that is greater-than-or-equal to every number in the set.
How do you define lim inf and lim sup?
Two additional remarks may be useful: The definition of lim inf is gotten from the definition of lim sup by simply reversing all inequalities. The definitions can be easily extended to ± ∞ in place of numbers t. Just adopt the convention that, even then, s refers to actual numbers, all of which are > − ∞ and < + ∞.
What is the difference between inferior limit and superior limit?
The superior limit is the larger of the two, and the inferior limit is the smaller of the two. The inferior and superior limits agree if and only if the sequence is convergent (i.e., when there is a single limit).
How are limits inferior/superior related to Big O notation?
Limits inferior/superior are related to big-O notation in that they bound a sequence only “in the limit”; the sequence may exceed the bound.
What does supremum and infimum mean?
The only thing I was able to pick out of it was the concept of infimum (greatest lower bound) and supremum (least upper bound), as… Stack Exchange Network