Limit (mathematics) - Limit As Standard Part

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,

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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:

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In this sense, taking the limit and taking the standard part are equivalent procedures.