Sequential Characterization of Integrability

A bounded function $f:[a,b]\to \mathbb{R}$ is Riemann integrable iff $\exists \set{P_n}$ a sequence of partitions, such that:

$$\lim_{n\to\infty}{ \Big[ U(f;P_n)- L(f;P_n) \Big] } =0$$

In that case:

$$\int_a^b f = \lim_{n\to\infty} U(f;P_n) = \lim_{n\to\infty} L(f;P_n)$$

Proof Hint

Cauchy criteria and squeeze theorem is used for both side proof.

For $\impliedby$: $$

• Consider the limit definition.
• Prove $f$ is Riemann integrable on $P_n$ by Cauchy criteria.
• Use squeeze theorem for $U(f;P_n)-U(f)\le U(f;P_n)-L(f;P_n)$ to prove limit of upper sum
• Prove limit of lower sum using the limit of upper sum

For $\implies$: Consider the below, where $n\in\mathbb{N}$.

$$0 \le U(f;P_n) - L(f;L_n) \le \frac{1}{n}$$

Theorem

Suppose $f$ is Riemann integrable on $[a,b]$.

$$\forall \epsilon \gt 0\; \exists \delta \gt 0\; \forall P\; \bigg( \lvert P \rvert \lt \delta \implies \Bigg\lvert \int_a^b f - \sum_{j=1}^n f(\zeta_j)I_j \Bigg\rvert \lt \epsilon \bigg)$$

where $\zeta_j \in [x_{j-1},x_j], j=1,2,\cdots,n$. $$

Proof Hint

$$\underline{ \int_a^b f } - \epsilon \; \lt \; L(f;P) \; \le \; \sum_{j=1}^n f(\zeta_j)I_j \; \le \; U(f;P) \; \lt \; \overline{ \int_a^b f } + \epsilon$$