Let \( \left[a, \ b \right] \) be any closed bounded interval. Any sequence of numbers \( x_0, \ x_1, \ x_2, \ x_3, \ \dots, \ x_n \) that satisfy \( a=x_0 < x_1 < x_2 < x_3 < \ \dots < x_n = b\) defines a partition of \( \left[a, \ b \right] \) and we write \( P = \{ x_0, \ x_1, \ x_2, \ x_3, \ \dots, \ x_n \}. \)

Essentially, a partition of \( \left[a, \ b \right] \) establishes a set of \( n \) subintervals \( \left[x_i, \ x_{i-1} \right] \) where \( 1 \le i \le n \). Equivalently, \( \left[a, \ b \right] = \bigcup_{i=1}^n \left[x_i, \ x_{i-1} \right] \)

I think that it is important to emphasize the following: If \( \Delta x = x_{i} – x_{i-1} \) is constant then the partition is called regular. The definition of a partition does not require that \( \Delta x \) be constant.

A partition \( R \) is called a refinement of a partition \( P \) if \( P \) is a subsequence of \( R \). That is \( P \subset R \).