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:
“An educational method that shall have liberty as its basis must intervene to help the child to a conquest of liberty. That is to say, his training must be such as shall help him to diminish as much as possible the social bonds which limit his activity.”
—Maria Montessori (1870–1952)
“When Freedom, from her mountain height,
Unfurled her standard to the air,
She tore the azure robe of night,
And set the stars of glory there;”
—Joseph Rodman Drake (1795–1820)
“Maria: You should get out of these clothes immediately. You’ll catch your death of pneumonia, you will.
Inspector Clouseau: Yes, yes, I probably will. But it’s all part of life’s rich pageantry.”
—Blake Edwards (b. 1922)