# Hyperfinite set

Hyperfinite sets can be used to approximate other sets. If a hyperfinite set approximates an interval, it is called a near interval with respect to that interval. Consider a hyperfinite set $K={k_{1},k_{2},\dots ,k_{n}}$ with a hypernatural n. K is a near interval for [a,b] if k1 = a and kn = b, and if the difference between successive elements of K is infinitesimal. Phrased otherwise, the requirement is that for every r ∈ [a,b] there is a kiK such that kir. This, for example, allows for an approximation to the unit circle, considered as the set $e^{i\theta }$ for θ in the interval [0,2π].
In terms of the ultrapower construction, the hyperreal line *R is defined as the collection of equivalence classes of sequences $\langle u_{n},n=1,2,\ldots \rangle$ of real numbers un. Namely, the equivalence class defines a hyperreal, denoted $[u_{n}]$ in Goldblatt's notation. Similarly, an arbitrary hyperfinite set in *R is of the form $[A_{n}]$ , and is defined by a sequence $\langle A_{n}\rangle$ of finite sets $A_{n}\subset \mathbb {R} ,n=1,2,\ldots$ 