Limit As Standard Part
In the context of a hyperreal enlargement of the number system, the limit of a sequence can be expressed as the standard part of the value of the natural extension of the sequence at an infinite hypernatural index . Thus,
- .
Here the standard part function "st" associates to each finite hyperreal, the unique finite real infinitely close to it (i.e., the difference between them is infinitesimal). This formalizes the natural intuition that for "very large" values of the index, the terms in the sequence are "very close" to the limit value of the sequence. Conversely, the standard part of a hyperreal represented in the ultrapower construction by a Cauchy sequence, is simply the limit of that sequence:
- .
In this sense, taking the limit and taking the standard part are equivalent procedures.
Read more about this topic: Limit (mathematics)
Famous quotes containing the words limit, standard and/or part:
“... there are two types of happiness and I have chosen that of the murderers. For I am happy. There was a time when I thought I had reached the limit of distress. Beyond that limit, there is a sterile and magnificent happiness.”
—Albert Camus (1913–1960)
“Any honest examination of the national life proves how far we are from the standard of human freedom with which we began. The recovery of this standard demands of everyone who loves this country a hard look at himself, for the greatest achievments must begin somewhere, and they always begin with the person. If we are not capable of this examination, we may yet become one of the most distinguished and monumental failures in the history of nations.”
—James Baldwin (1924–1987)
“Good God! how often are we to die before we go quite off this stage? In every friend we lose a part of ourselves, and the best part.”
—Alexander Pope (1688–1744)