EXERCISES 1A
7.
Solution.
By definition of supremum
\[
L \left( f, \left[ a,b \right] \right) \geq L \left( f, P_n, \left[ a,b \right] \right)
\]
Then by preservation of non strict inequalities
\[
L \left( f, \left[ a,b \right] \right) \geq \lim_{ n \to \infty} L \left( f, P_n, \left[ a,b \right] \right)
\]
Let \(P_n\) be denoted by intervals \(I_j = \left[ x_{j-1}, x_j \right]\) where \(x_j = a + j \frac{b-a}{2^j}\) for \(j \in \left\{ 1,\dots, 2^n \right\}\)
Let \(R\) be an arbitrary partition of \([a,b]\), with endpoints \(\left\{ a=r_0, \dots, r_m=b \right\}\) and \(R_j = [r_{j-1}, r_j]\).
We can then make \(n\) sufficently large such that every \(P_n\) interval \(I_j\) is either fully within some \(R\) interval or lives exactly within two adjacent \(R_k, R_{k+1}\),
i.e. \(x_{j-1} < r_k < x_j\) for some \(k \in \left\{ 1, \dots, m \right\}\) and \(j \in \left\{ 1, \dots, 2^n \right\}\)
\[
\begin{align*}
& L(f, P_n, [a,b]) - L(f, R, [a,b]) =
\\
&
\sum_{j : I_j \subseteq R_k \text{ some } k \in \left\{ 1, \dots,m \right\}}
\left| I_j \right|
\left(
\inf_{I_j}{f} - \inf_{R_k}{f}
\right)
\\ &+
\sum_{j : x_{j-1} < r_k < x_j \text{ some } k \in \left\{ 1, \dots,m \right\}}
\left| I_j \cap R_k \right|
\left(
\inf_{I_j}{f} - \inf_{R_k}{f}
\right)
+ \left| I \cap R_{k+1} \right|
\left(
\inf_{I_j}{f} - \inf_{R_{k+1}}{f}
\right)
\end{align*}
\]
Now, for \(j: x_{j-1} < r_k < x_j \text{ some } k \in \left\{ 1, \dots,m \right\}\)
\[
\begin{align*}
&
\left|
\left| I_j \cap R_k \right|
\left(
\inf_{I_j}{f} - \inf_{R_k}{f}
\right)
+ \left| I \cap R_{k+1} \right|
\left(
\inf_{I_j}{f} - \inf_{R_{k+1}}{f}
\right)
\right|
\\ &=
\left|
\left( r_k - x_{j-1} \right)
\left(
\inf_{I_j}{f} - \inf_{R_k}{f}
\right)
+ \left( x_j - r_k \right)
\left(
\inf_{I_j}{f} - \inf_{R_{k+1}}{f}
\right)
\right|
\\ &\leq
(a-b) 2^{-n} 2c + (a-b) 2^{-n} 2c
\end{align*}
\]
Where \(c \geq \left| f(x) \right| \forall x \in [a,b]\) is the \(f\) bound.
Now the second sum over \(\left\{ j: x_{j-1} < r_k < x_j \text{ some } k \in \left\{ 1, \dots,m \right\} \right\}\) has at most \(m\) elments since there are at most \(m\) endpoints between two \(R\) intervals for each \(I_j\) to overlap with.
Therefore
\[
\left|
\sum_{j : x_{j-1} < r_k < x_j \text{ some } k \in \left\{ 1, \dots,m \right\}}
\left| I_j \cap R_k \right|
\left(
\inf_{I_j}{f} - \inf_{R_k}{f}
\right)
+ \left| I \cap R_{k+1} \right|
\left(
\inf_{I_j}{f} - \inf_{R_{k+1}}{f}
\right)
\right|
\leq 4 c m (b-a)2^{-n}
\]
Which gives
\[
\begin{align*}
L(f, P_n, [a,b]) - L(f, R, [a,b])
& \geq
\sum_{j : I_j \subseteq R_k \text{ some } k \in \left\{ 1, \dots,m \right\}}
\left| I_j \right|
\left(
\inf_{I_j}{f} - \inf_{R_k}{f}
\right)
&- 4cm (b-a)^{-n}
\\
&\geq
-4cm (b-a)^{-n}
\end{align*}
\]
Taking limits gives
\[
\lim_{n \to \infty} L(f, P_n, [a,b]) - L(f, R, [a,b]) \geq 0
\]