Partition of an interval

In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x0, x1, x2, …, xn of real numbers such that In other terms, a partition of a compact interval I is a strictly increasing sequence of numbers (belonging to the interval I itself) starting from the initial point of I and arriving at the final point of I.

Specifically, as finer partitions of a given interval are considered, their mesh approaches zero and the Riemann sum based on a given partition approaches the Riemann integral.

It is possible to define a partial order on the set of all tagged partitions by saying that one tagged partition is bigger than another if the bigger one is a refinement of the smaller one.

[citation needed] Suppose that x0, …, xn together with t0, …, tn − 1 is a tagged partition of [a, b], and that y0, …, ym together with s0, …, sm − 1 is another tagged partition of [a, b].

We say that y0, …, ym together with s0, …, sm − 1 is a refinement of a tagged partition x0, …, xn together with t0, …, tn − 1 if for each integer i with 0 ≤ i ≤ n, there is an integer r(i) such that xi = yr(i) and such that ti = sj for some j with r(i) ≤ j ≤ r(i + 1) − 1.