Let f:[a, b] →ℝ be a bounded function. Without using Lemma...

Let $f:[a, b] \rightarrow \mathbb{R}$ be a bounded function. Without using Lemma $6.30$, show that $f$ is Riemann integrable if and only if there is $r \in \mathbb{R}$ satisfying the following condition: Given $\epsilon>0$, there is a partition $P_{\epsilon}$ of $[a, b]$ such that $|S(P, f)-r|<\epsilon$, where $P$ is any refinement of $P_{\epsilon}$ and $S(P, f)$ is any Riemann sum for $f$ corresponding to $P$.

Key Concepts

Uniform Convergence of Riemann Sums

This concept pertains to the condition where all Riemann sums, taken over any refinement of a given partition, uniformly approximate a particular value within any prescribed tolerance. This uniformity ensures that regardless of the specific choice of sample points within each subinterval, the resulting sums remain close to the target value. It is an essential ingredient in establishing the equivalence between the conventional definition of integrability and the condition that every refinement of a partition yields Riemann sums near a fixed number.

Riemann Sums

A Riemann sum is an approximation to the integral of a function, obtained by selecting a sample point in each subinterval of a partition and forming a sum of the product of function values at these points and the corresponding subinterval lengths. The idea is that as the subintervals become smaller through refinements, these sums should approach the actual area under the curve if the function is Riemann integrable.

Riemann Integrability

This is the concept of determining whether a bounded function on a closed interval has a well-defined area under its graph. It involves showing that the upper and lower approximations (using the supremum and infimum over subintervals) can be made arbitrarily close, meaning that as the partitions become finer, the difference between these approximations approaches zero. This is the foundational idea behind the Riemann integral, indicating that the 'limit' of the sums exists and is unique.

Partitions and Refinements

Partitions involve dividing an interval into smaller subintervals, which serve as the basis for approximating the area under a function's graph. A refinement of a partition is simply a further subdivision that includes all the original points and possibly more. The concept is crucial because any properties proven for a coarser partition need to extend to all its refinements in order to establish uniformity in the approximation process.

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