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An infimum of a set is always and only defined relative to a superset of the set in question. For example, there is no infimum of the positive real numbers inside the positive real numbers (as their own superset), nor any infimum of the positive real numbers inside the complex numbers with positive real part.
In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers.Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the ...
Exactly in the same way one defines the essential infimum as the supremum of the essential lower bound s, that is, = {: ({: <}) =} if the set of essential lower bounds is nonempty, and as otherwise; again there is an alternative expression as = {: ()} (with this being if the set is empty).
A real number x is called an upper bound for S if x ≥ s for all s ∈ S. A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S. The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real ...
In the extended real numbers every set has a supremum (resp. infimum) which of course may be (resp. ) if the set is unbounded. An important use of the extended reals is that any set of non negative numbers a i ≥ 0 , i ∈ I {\displaystyle a_{i}\geq 0,i\in I} has a well defined summation order independent sum
An interval is a subset of the real numbers that contains all real numbers lying between any two numbers of the subset. The endpoints of an interval are its supremum, and its infimum, if they exist as real numbers. [1] If the infimum does not exist, one says often that the corresponding endpoint is .
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Define [1] = {: =} and = {: =}, where the expressions inside the brackets on the right are, respectively, the limit infimum and limit supremum of the real-valued sequence (). Again, if these two sets are equal, then the set-theoretic limit of the sequence A n {\displaystyle A_{n}} exists and is equal to that common set, and either set as ...