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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.
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 ...
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 .
For example, the ancient Babylonians discovered a method for computing square roots of numbers. In contrast, the famed Archimedes constructed sequences of polygons, that inscribed and circumscribed a unit circle , in order to get a lower and upper bound for the circles circumference - which is the circle number Pi ( π {\displaystyle \pi } ).
The supremum of B is then equal to the infimum of X: since each element of X is an upper bound of B, sup B is smaller than all elements of X, i.e. sup B is in B. It is the greatest element of B and hence the infimum of X. In a dual way, the existence of all infima implies the existence of all suprema.
The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well. [ 1 ] [ 4 ] The pointwise limit of a sequence of measurable functions f n : X → Y {\displaystyle f_{n}:X\to Y} is measurable, where Y {\displaystyle Y} is a metric space ...
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).
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 ...