Let f:[a, b] →ℝ be integrable. Show that the Riemann...

Let $f:[a, b] \rightarrow \mathbb{R}$ be integrable. Show that the Riemann integral of $f$ is the unique real number $r$ satisfying the following condition: For every $\epsilon>0$, there is a partition $P_{\epsilon}$ of $[a, b]$ such that $$ r-\epsilon<L\left(P_{\epsilon}, f\right) \leq r \leq U\left(P_{\epsilon}, f\right)<r+\epsilon $$

Let $f:[a, b] \rightarrow \mathbb{R}$ be integrable. Show that the Riemann integral of $f$ is the unique real number $r$ satisfying the following condition: For every $\epsilon>0$, there is a partition $P_{\epsilon}$ of $[a, b]$ such that
$$
r-\epsilon<L\left(P_{\epsilon}, f\right) \leq r \leq U\left(P_{\epsilon}, f\right)<r+\epsilon
$$

Key Concepts

Riemann Integral

The Riemann integral is a method of defining the integral of a function by approximating the area under its curve as the limit of sums of products of function values and subinterval lengths. It provides a formal definition for the concept of accumulation (such as area or total change) and requires that as the partition of the interval becomes finer, the lower and upper sums converge to a unique value.

Partitions

A partition of a closed interval is a finite sequence of points that divide the interval into subintervals. Partitions are central in the process of Riemann integration because they determine how the interval is broken down for the approximation of the area under the curve. The accuracy of the integral approximation increases as the maximum subinterval length (mesh size) of the partition decreases.

Lower Sum and Upper Sum

The lower sum is the sum calculated by taking the infimum (the greatest lower bound) of the function on each subinterval multiplied by the length of that subinterval, while the upper sum is obtained by using the supremum (the least upper bound) of the function on these subintervals. These sums serve as bounds for the integral, and their convergence to the same value as the partition is refined is both a necessary and sufficient condition for the function to be Riemann integrable.

Epsilon Criterion for Integration

The epsilon criterion formalizes the idea of convergence in the context of Riemann integration by stating that for every positive epsilon, there exists a partition such that the lower and upper sums differ from the integral by less than epsilon. This condition ensures that the approximations can be made arbitrarily close to the actual value of the integral, thereby establishing the uniqueness of the Riemann integral.

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