Partition Points

Given a closed real interval [a, b], a finite partition is formed by choosing a positive integer n and dividing [a, b] into n equal parts, as in Figure 3.8.3. Each part will be a subinterval of length t = (b - a)/n. The n subintervals are

[a, a + t], [a + t, a + 2t],... ,[a + (n - 1)t, b].

Figure 3.8.3

The endpoints

a, a + t, a + 2t, ..., a + (n - 1)t, a + nt = b

are called partition points.

The real interval [a, b] is contained in the hyperreal interval [a, b]*, which is the set of all hyperreal numbers x such that a ≤ x ≤ b. An infinite partition is applied to the hyperreal interval [a, b]* rather than the real interval. To form an infinite partition of [a, b]*, choose a positive infinite hyperinteger H and divide [a, b]* into H equal parts as shown in Figure 3.8.4. Each subinterval will have the same infinitesimal length δ = (b - a)/H. The H subintervals are

[a, a + δ], [a + δ, a + 2δ],..., [a + (K - 1)δ, a + Kδ],..., [a + (H - 1)δ, b],

and the partition points are

a, a + δ, a + 2δ,... , a + Kδ,... , a + Hδ = b,

where K runs over the hyperintegers from 1 to H. Every hyperreal number x between a and b belongs to one of the infinitesimal subintervals,

a + (K- 1)δ ≤ x < a + Kδ.

Figure 3.8.4