
Medium
Using any equivalent definition of integration
(see Question 4), show that integration is a linear operator; that, show that
if \(f_1\) and \(f_1\) are integrable on \([a,b]\), then \(c_1 f_1+ c_2f_2\) is
integrable on \([a,b]\) and \[ \int_a^b \left[ c_1f_1(x) + c_2f_2(x) \right]dx
= c_1 \int_a^b f_1(x) dx + c_2 \int_a^b f_2(x) dx. \]

Medium
Show that the set \(S = \left\{\frac{1}{n}\right\}_{n=1}^\infty\subseteq \mathbb R\) has Jordan measure zero.

Medium
More generally, show that if \((a_n)_{n=1}^\infty\) is any convergent sequence, then \((a_n)\) has Jordan measure zero.

Hard
Prove that the following statements
are equivalent:
 The bounded function \(f: [a,b] \to \mathbb R\)
satisfies the following: there is a real number \(I\) such that, for all
\(\epsilon >0\) there exists a \(\delta>0\) having the property that
whenever \(P \in \mathcal P_{[a,b]}\) is a partition of \([a,b]\) with
\(\ell(P) < \delta\), then \[ S(f,P)  I < \epsilon. \]
 If \(f:[a,b] \to \mathbb R\) is a bounded function, then \[
\sup_{P\in\mathcal P_{[a,b]}} L_f(P) = \inf_{P\in\mathcal
P_{[a,b]}} U_f(P). \]
 The bounded function \(f:[a,b] \to \mathbb R\) satisfies the
following property: For every \(\epsilon >0\) there exists a
partition \(P\in\mathcal P_{[a,b]}\) such that \[ U_f(P)  L_f(P) <
\epsilon. \]
 The bounded function \(f:[a,b] \to \mathbb R\) satisfies the
property that for every \(\epsilon >0\), there exists a \(\delta
>0\) such that whenever \(P,Q \in \mathcal P_{[a,b]}\) are two
partitions satisfying \(\ell(P)<\delta, \ell(Q)<\delta\) then \[
S(f,P)  S(f,Q) < \epsilon. \]

Hard
Show that every Riemann integrable function \(f:[a,b] \to \mathbb R\)
is bounded; that is, there exists some \(M>0\) such that \(f(x)\leq M\)
for all \(x \in [a,b]\). (NOTE: feel free to SKIP this question, as some of the lectures defined Riemann integrable over closed rectangles only for bounded functions to begin with)

Hard
Show that if \(f:[a,b] \to \mathbb R\) is Riemann integrable, then
\(F(x) = \int_a^x f(s) ds\) is uniformly continuous on \([a,b]\). [Hint:
Use the fact that integrable functions are bounded, together with a couple of properties for integrals, one involving an absolute value]