Sum of Integrals on Adjacent Intervals for Integrable Functions/Lemma/Proof 2
Theorem
Let $\closedint a b$ be a closed real interval.
Let $c$ be a real number.
Let $a < c < b$.
Let $f$ be a real function defined on $\closedint a b$.
Let $\map L S$ be the lower Darboux sum of $f$ on $\closedint a b$ where $S$ is a subdivision of $\closedint a b$.
Let $P$ and $Q$ be finite subdivisions of $\closedint a b$.
Let:
- $Q = P \cup \set c$.
Then:
- $\map L Q \ge \map L P$
Proof
Let $P = \set {x_0, x_1, \ldots, x_n}$.
Suppose that:
- $c \in P$
Then:
- $Q = P$
We have:
- $\map L P \ge \map L P$
- $\leadsto \map L Q \ge \map L P$ as $Q = P$
This finishes the proof for this case.
The only other possibility for $c$ is:
- $x_{j-1} < c < x_j$
where $1 \le j \le n$.
Let $m_i$ be the infimum of $f$ on the interval $\closedint {x_{i - 1}} {x_i}$ for all $i \in \set {1, 2, \ldots, n}$.
We have:
\(\ds \map L P\) | \(=\) | \(\ds \sum_{i \mathop = 1}^n m_i \paren {x_i - x_{i - 1} }\) | Definition of Lower Darboux Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^{j - 1} m_i \paren {x_i - x_{i - 1} } + m_j \paren {x_j - x_{j - 1} } + \sum_{i \mathop = j + 1}^n m_i \paren {x_i - x_{i - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^{j - 1} m_i \paren {x_i - x_{i - 1} } + m_j \paren {x_j − c + c - x_{j - 1} } + \sum_{i \mathop = j + 1}^n m_i \paren {x_i - x_{i - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^{j - 1} m_i \paren {x_i - x_{i - 1} } + m_j \paren {x_j − c} + m_j \paren {c - x_{j - 1} } + \sum_{i \mathop = j + 1}^n m_i \paren {x_i - x_{i - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^{j - 1} m_i \paren {x_i - x_{i - 1} } + m_j \paren {c - x_{j - 1} } + m_j \paren {x_j − c} + \sum_{i \mathop = j + 1}^n m_i \paren {x_i - x_{i - 1} }\) |
Let $r$ be the infimum of $f$ on the interval $\closedint {x_{j - 1}} c$.
Let $s$ be the infimum of $f$ on the interval $\closedint c {x_j}$.
The interval $\closedint {x_{j - 1}} c$ is a subset of $\closedint {x_{j - 1}} {x_j}$.
Therefore, a lower bound of $f$ on the interval $\closedint {x_{j - 1}} {x_j}$ is a lower bound of $f$ on the interval $\closedint {x_{j - 1}} c$ as well.
The infimum of $f$ on the interval $\closedint {x_{j - 1}} {x_j}$ is a lower bound for $f$ on this interval.
Accordingly, the infimum of $f$ on the interval $\closedint {x_{j - 1}} {x_j}$ is a lower bound of $f$ on the interval $\closedint {x_{j - 1}} c$ as well.
Therefore, the infimum of $f$ on the interval $\closedint {x_{j - 1}} c$ is greater than or equal to the infimum of $f$ on the interval $\closedint {x_{j - 1}} {x_j}$.
In other words:
- $r \ge m_j$
A similar analysis of $\closedint c {x_j}$ gives:
- $s \ge m_j$
We find:
\(\ds \map L P\) | \(=\) | \(\ds \sum_{i \mathop = 1}^{j - 1} m_i \paren {x_i - x_{i - 1} } + m_j \paren {c - x_{j - 1} } + m_j \paren {x_j − c} + \sum_{i \mathop = j + 1}^n m_i \paren {x_i - x_{i - 1} }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{i \mathop = 1}^{j - 1} m_i \paren {x_i - x_{i - 1} } + r \paren {c - x_{j - 1} } + s \paren {x_j − c} + \sum_{i \mathop = j + 1}^n m_i \paren {x_i - x_{i - 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map L Q\) | Definition of Lower Darboux Sum |
This finishes the proof.
$\blacksquare$