# Problem Set 14

1. 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 $\int_a^b \left[ c_1f_1(x) + c_2f_2(x) \right]dx = c_1 \int_a^b f_1(x) dx + c_2 \int_a^b f_2(x) dx.$

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

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

4. Hard
Prove that the following statements are equivalent:
1. 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 $|S(f,P) - I| < \epsilon.$
2. If $$f:[a,b] \to \mathbb R$$ is a bounded function, then $\sup_{P\in\mathcal P_{[a,b]}} L_f(P) = \inf_{P\in\mathcal P_{[a,b]}} U_f(P).$
3. 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 $U_f(P) - L_f(P) < \epsilon.$
4. 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 $|S(f,P) - S(f,Q)| < \epsilon.$

5. 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)

6. 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]

If you have selected no questions, the "Download Problem Set" button will default to sending the entire problem set, without any solutions.