Partitions of a Closed Interval

Definition: Let $I = [a, b]$ be a closed interval. A Partition is a set $P = \{ a = x_0, x_1, ..., x_n = b \}$ that satisfies the inequalities $a = x_0 < x_1 < ... < x_n = b$. The closed subintervals of $[a, b]$ corresponding to the partition $P$ are $[x_0, x_1], [x_1, x_2], ..., [x_{n-1}, x_n]$ is called a Partitioning or Subdivision of $[a, b]$. The set of all partitions $P$ of $I$ is denoted $\mathscr{P}[a, b]$.

The term "partition" can be used interchangeably to denote the set of points forming the partitioning or as the partitioning itself.

For example, consider the interval $[0, 6]$. One such partition is the set $P = \{ 0, 1, 2, 3, 4, 5, 6 \}$ and the corresponding partitioning of $[0, 6]$ into subintervals from $P$ is

(1)

\begin{align} \quad [0,1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6] \end{align}

Of course, $Q = \{0, 3, 6 \}$ is also a partition of $[0, 6]$ and the corresponding partitioning of $[0, 6]$ into subintervals from $Q$ is:

(2)

\begin{align} \quad [0, 3], [3, 6] \end{align}

A partition need not form subintervals that have equal length. For example, $R = \{ 0, 2, 3, 4.5, 6 \}$ is a partition on $[0, 6]$ into the subintervals:

(3)

\begin{align} \quad [0, 2], [2, 3], [3, 4.5], [4.5, 6] \end{align}

As you can see, the set $\mathscr{P}[a, b]$ contains many partitions whose subintervals have varying lengths. For each partition $P = \{ a = x_0, x_1, ..., x_n \} \in \mathscr{P}[a, b]$ it is customary for each $k \in \{ 1, 2, ..., n \}$ to write the length of the $k^{\mathrm{th}}$ subinterval $[x_{k-1}, x_{k}]$ as:

(4)

\begin{align} \quad \Delta x_k = x_k - x_{k-1} \end{align}

We will use this notation extensively later on.

Furthermore, the sum of the lengths of all $n$ subintervals of the partition $P \in \mathscr{P}[a, b]$ can be readily verified to equal the length $b - a$ of the entire interval $[a, b]$ since:

(5)

\begin{align} \quad \sum_{k=1}^{n} \Delta x_k = \sum_{k=1}^{n} [x_k - x_{k-1}] = [x_1 - x_0] + [x_2 - x_1] + ... + [x_n - x_{n-1}] = x_n - x_0 = b - a \end{align}

This fact will also be important later on.