Question

Prove: If \(f\) is integrable on \([a, b]\) and \(\epsilon>0,\) then \(S(P)-s(P)<\epsilon\) if \(\|P\|\) is sufficiently small. HINT: Use Theorem 3.1 .4

Short answer

Question: Prove that if a function \(f\) is integrable on the interval \([a, b]\) and \(\epsilon > 0\), then the difference between the upper sum and the lower sum, \(S(P) - s(P)\), is less than \(\epsilon\) if the norm of the partition, \(\|P\|\), is sufficiently small.

Step-by-step solution

Define the given information

We are given that \(f\) is an integrable function on the interval \([a, b]\) and \(\epsilon > 0\). Our goal is to show that \(S(P)-s(P)<\epsilon\) if \(\|P\|\) is sufficiently small.

Review Theorem 3.1.4 and apply it to the given function

Recall Theorem 3.1.4: A function \(f\) is integrable on \([a, b]\) if and only if for every \(\epsilon > 0\), there exists a partition \(P\) of \([a, b]\) such that \(U(P,f)- L(P,f) < \epsilon\). Since \(f\) is integrable on \([a, b]\), we know that there exists a partition \(P\) such that the difference between the upper sum \(U(P,f)\) and the lower sum \(L(P,f)\) is less than \(\epsilon\).

Refine the partition to ensure that \(\|P\|\) is small enough

Now, we need to find a partition \(P\) such that not only does \(U(P,f)- L(P,f) < \epsilon\) hold true, but also \(\|P\|\) is sufficiently small. To do this, we can first take the partition \(P\) for which \(U(P,f)- L(P,f) < \epsilon\), and then refine it by adding more points to the partition to make it smaller. Note that refining a partition will never increase the difference between the upper and lower sums. Therefore, the refined partition will also have a difference between the upper and lower sums that is smaller than \(\epsilon\). Let's denote this refined partition as \(P'\), such that \(\|P'\|\) is sufficiently small.

Show that \(S(P')-s(P')

Using the refined partition \(P'\), we can now compute the upper and lower sums for \(f\) on \([a, b]\), denoted by \(S(P')\) and \(s(P')\) respectively. Recall that \(S(P)\) and \(s(P)\) are defined as \(S(P) = \sum_{i=1}^{n} M_i \Delta x_i\) and \(s(P) = \sum_{i=1}^{n} m_i \Delta x_i\), where \(M_i\) and \(m_i\) are the maximum and minimum values of \(f\) on \([x_{i-1}, x_i]\) respectively. Since we chose the refined partition \(P'\) such that the difference between the upper and lower sums is smaller than \(\epsilon\), we can conclude that \(S(P')-s(P') < \epsilon\). Hence, we have proved the statement: If \(f\) is integrable on \([a, b]\) and \(\epsilon>0\), then \(S(P)-s(P)<\epsilon\) if \(\|P\|\) is sufficiently small.

Key concepts

Real Analysis

Real analysis is a branch of mathematics dealing with the behavior and properties of real numbers, sequences, series, and real-valued functions. This field underpins a vast array of higher mathematics, playing a crucial role in both pure and applied sciences. In the context of integrable functions, real analysis provides the framework for understanding the calculation of area under curves, convergence of functions, and the intricacies of continuity and limits. The concept of an integrable function, which comes from studying the area under a curve, stems from defining and approximating areas using simpler geometric shapes.Real analysis uses rigorous proofs to establish foundational results such as the convergence criteria for sequences and series, and the Bolzano-Weierstrass theorem, which states that every bounded sequence has a convergent subsequence. Understanding these principles is vital for grasping more advanced topics in real analysis, like the one we encounter when discussing the integrability of a function over a certain interval.

Upper and Lower Sums

In real analysis, upper and lower sums are vital concepts used to define the integral of a function. They provide a way to approximate the area under a curve by summing up the areas of rectangles that either overestimate or underestimate the actual area. The upper sum, denoted by \(U(P,f)\), is the sum of the areas of the rectangles formed by taking the maximum value of the function on each subinterval of a partition. Conversely, the lower sum, \(L(P,f)\), uses the minimum value of the function on each subinterval.

How Are They Calculated?

The process starts with a partition \(P\) of the interval. For the upper sum, we find the maximum function value on each subinterval and multiply it by the subinterval's length, effectively 'filling in' any gaps below the curve. The lower sum does the opposite, multiplying the minimum function values by the lengths of the subintervals, leaving 'gaps' above the curve. The narrower the subintervals, the closer these sums are to the actual integral, which leads to the important concept of a function being integrable if the upper and lower sums can be made arbitrarily close.

Partition of an Interval

A partition of an interval in the scope of real analysis is the division of a closed interval \[a, b\] into smaller subintervals that cover the entire interval without overlapping. Formally, a partition \(P\) of an interval \[a, b\] can be expressed as a finite sequence of points \( a = x_0 < x_1 < x_2 < ... < x_{n-1} < x_n = b \). Each consecutive pair of points \( (x_{i-1}, x_i)\) defines a subinterval of the partition.

Importance of Partition Refinement for Integrability

The beauty of partitions lies in their adaptability; by refining a partition (adding more points), one can create a more accurate approximation of the integral. A function \(f\) is said to be integrable if we can make the difference between the upper and lower sums arbitrarily small for some choice of partition with sufficiently fine subintervals. This is precisely the notion we employ when proving certain properties of integrable functions, relying on the idea that a sufficiently refined partition will reduce the discrepancy between these sums, thus solidifying the connection between the partitions and integral estimations.

Epsilon-Delta Definition of Integrability

The epsilon-delta definition is a formalization used extensively in real analysis to define limits, continuity, and, as pertinent to our discussion, integrability. For a function to be integrable on an interval, the epsilon-delta criterion demands that for every \(\epsilon > 0\) (no matter how small), there must exist a partition \(P\) of the interval such that the difference between the upper sum \(U(P,f)\) and the lower sum \(L(P,f)\) is less than \(\epsilon\), i.e., \(|U(P,f) - L(P,f)| < \epsilon\).

Understanding the Criterion Through an Example

To apply this definition, consider the function \(f\) integrable if for any tiny positive \(\epsilon\), we can find a delta (expressed here as the 'norm' of a partition \(||P||\), which is the length of the largest subinterval) such that when \(||P||\) is less than delta, the aforementioned condition on the upper and lower sums is met. This provides a practical method to prove that a function is integrable by demonstrating that no matter how small the \(\epsilon\), we can always find a partition fine enough (with small enough \(||P||\)) to satisfy the integral approximation within the desired accuracy.