Limit of A Sequence - Definition in Hyperreal Numbers

Definition in Hyperreal Numbers

The definition of the limit using the hyperreal numbers formalizes the intuition that for a "very large" value of the index, the corresponding term is "very close" to the limit. More precisely, a real sequence tends to L if for every infinite hypernatural H, the term x_H is infinitely close to L, i.e., the difference x_H - L is infinitesimal. Equivalently, L is the standard part of x_H

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Thus, the limit can be defined by the formula

where the limit exists if and only if the righthand side is independent of the choice of an infinite H.