Limit of Sequence
Given a sequence of real numbers, if we say L is the limit of the sequence and write
if for every infinite hypernatural n, we have st(xn)=L (here the extension principle is used to define xn for every hyperinteger n).
This definition has no quantifier alternations.The standard (ε, δ)-style definition on the other hand does have quantifier alternations:
Read more about this topic: Non-standard Calculus
Famous quotes containing the words limit and/or sequence:
“Today, the notion of progress in a single line without goal or limit seems perhaps the most parochial notion of a very parochial century.”
—Lewis Mumford (1895–1990)
“We have defined a story as a narrative of events arranged in their time-sequence. A plot is also a narrative of events, the emphasis falling on causality. “The king died and then the queen died” is a story. “The king died, and then the queen died of grief” is a plot. The time sequence is preserved, but the sense of causality overshadows it.”
—E.M. (Edward Morgan)