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.
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Famous quotes containing the words limit, standard and/or part:
“Can you find out the deep things of God? Can you find out the limit of the Almighty?”
—Bible: Hebrew, Job 11:7.
“If the Revolution has the right to destroy bridges and art monuments whenever necessary, it will stop still less from laying its hand on any tendency in art which, no matter how great its achievement in form, threatens to disintegrate the revolutionary environment or to arouse the internal forces of the Revolution, that is, the proletariat, the peasantry and the intelligentsia, to a hostile opposition to one another. Our standard is, clearly, political, imperative and intolerant.”
—Leon Trotsky (1879–1940)
“Funny ain’t it. Here I am worrying about a woman. Men don’t worry much about women when they’re around. But when it gets way off from home like we are now, and where he knows he’s going a lot further away ... I mean that’s when a woman gets workin’ in your mind. You reckon you’re a fool for not noticin’ before how, how big a part of things they be. There ain’t nothin’ like seein’ a woman’s face.”
—Dudley Nichols (1895–1960)