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 of, limit and/or sequence:
“Berowne they call him, but a merrier man,
Within the limit of becoming mirth,
I never spent an hour’s talk withal.”
—William Shakespeare (1564–1616)
“We are rarely able to interact only with folks like ourselves, who think as we do. No matter how much some of us deny this reality and long for the safety and familiarity of sameness, inclusive ways of knowing and living offer us the only true way to emancipate ourselves from the divisions that limit our minds and imaginations.”
—bell hooks (b. 1955)
“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)