The Case of Sequences of Real Numbers
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 complete totally ordered set, which is a complete lattice.
Read more about this topic: Limit Superior And Limit Inferior
Famous quotes containing the words case, real and/or numbers:
“In the case of a deliberate, palpable, and dangerous exercise of ... powers not granted by the compact, the States ... are in duty bound to interpose for arresting the progress of the evil, and for maintaining within their respective limits the authorities, rights, and liberties appertaining to them.”
—James Madison (1751–1836)
“Fear can supplant our real problems only to the extent—unwilling either to assimilate or to exhaust it—we perpetuate it within ourselves like a temptation and enthrone it at the very heart of our solitude.”
—E.M. Cioran (b. 1911)
“He bundles every forkful in its place,
And tags and numbers it for future reference,
So he can find and easily dislodge it
In the unloading. Silas does that well.
He takes it out in bunches like birds’ nests.”
—Robert Frost (1874–1963)