Problem 10 - A Course in Calculus and Real Analysis

Question

Let \(f:[a, b] \rightarrow \mathbb{R}\) be any function. Suppose there is \(r \in \mathbb{R}\) and for each \(n \in \mathbb{N}\), there are integrable functions \(g_{n}, h_{n}:[a, b] \rightarrow \mathbb{R}\) with \(g_{n} \leq f \leq h_{n}\) such that \(\int_{a}^{b} g_{n}(x) d x \rightarrow r\) and \(\int_{a}^{b} h_{n}(x) d x \rightarrow r\) as \(n \rightarrow \infty\). Show that \(f\) is integrable and the Riemann integral of \(f\) is equal to \(r\).

Step-by-Step Solution

Answer

The function \(f\) is Riemann integrable because we can find a partition \(P\) such that \(U(P,f) - L(P,f) < \epsilon\) for any \(\epsilon > 0\). Moreover, the Riemann integral of \(f\) is equal to \(r\), as we have shown that \(|\int_{a}^{b} f(x) dx - r| < \frac{\epsilon}{3}\) for every \(\epsilon > 0\).

Step 1: Epsilon-N Lemma

First, we need to demonstrate that given \(\epsilon > 0\), there exists an \(N\) such that for all \(n \geq N\), \( |\int_{a}^{b} g_{n}(x) dx - r| < \frac{\epsilon}{3} \) and \( | \int_{a}^{b} h_{n}(x) dx - r| < \frac{\epsilon}{3} \). Since \(\int_{a}^{b} g_{n}(x) dx \rightarrow r\) and \(\int_{a}^{b} h_{n}(x) dx \rightarrow r\) as \(n \rightarrow \infty\), we can use the epsilon-N lemma to find such an \(N\) that satisfies the conditions above.

Step 2: Create a common partition for \(g_{N}\), \(h_{N}\) and \(f\)

Next, we need to find a common partition \(P\) for the functions \(g_{N}\), \(h_{N}\), and \(f\). Since \(g_{N}\) and \(h_{N}\) are integrable, for each of them there exists a partition such that the difference between the upper sum and lower sum is less than \(\frac{\epsilon}{3}\). Let \(P_{g_{N}}\) and \(P_{h_{N}}\) be such partitions for \(g_{N}\) and \(h_{N}\) respectively. Then, define the partition \(P\) for \(f\) as the union of \(P_{g_{N}}\) and \(P_{h_{N}}\).

Step 3: Verify conditions of Riemann integral definition

Now we will verify that the conditions of the Riemann integral definition are met for the function \(f\) on the partition \(P\). In particular, we need to show that there is a partition \(P\) such that: \( U(P,f) - L(P,f) < \epsilon \) First, let's compute the difference between the upper sum and lower sum of \(f\) over the partition \(P\). Since \(g_{N} \leq f \leq h_{N}\), we have: \( U(P,f) - L(P,f) \leq U(P,h_{N}) - L(P,g_{N}) \) Moreover, since the original selected partitions \(P_{g_{N}}\) and \(P_{h_{N}}\) for \(g_{N}\) and \(h_{N}\): \( U(P_{g_{N}},g_{N}) - L(P_{g_{N}},g_{N}) < \frac{\epsilon}{3} \) and \( U(P_{h_{N}},h_{N}) - L(P_{h_{N}},h_{N}) < \frac{\epsilon}{3} \) According to the properties of upper and lower sums, the differences above can only increase when we change the partition from \(P_{g_{N}}\) to \(P\) and from \(P_{h_{N}}\) to \(P\). Hence, we get: \( U(P,h_{N}) - L(P,g_{N}) \leq U(P_{h_{N}},h_{N}) - L(P_{g_{N}},g_{N}) \leq \frac{2\epsilon}{3} \) Therefore, we have shown that there exists a partition \(P\) such that: \( U(P,f) - L(P,f) \leq U(P,h_{N}) - L(P,g_{N}) \leq \frac{2\epsilon}{3} < \epsilon \) This implies that the function \(f\) is integrable on the interval \([a, b]\).

Step 4: Show Riemann integral of \(f\) equals \(r\)

To conclude the solution, we will demonstrate that the Riemann integral of \(f\) is equal to \(r\). Since \(g_{N} \leq f \leq h_{N}\), we have: \( \int_{a}^{b} g_{N}(x) dx \leq \int_{a}^{b} f(x) dx \leq \int_{a}^{b} h_{N}(x) dx \) From the epsilon-N lemma in Step 1, we know that for \(n \geq N\): \( | \int_{a}^{b} g_{n}(x) dx - r | < \frac{\epsilon}{3} \) and \( | \int_{a}^{b} h_{n}(x) dx - r | < \frac{\epsilon}{3} \) Therefore, we have: \( r - \frac{\epsilon}{3} < \int_{a}^{b} g_{N}(x) dx \leq \int_{a}^{b} f(x) dx \leq \int_{a}^{b} h_{N}(x) dx < r + \frac{\epsilon}{3} \) From the properties of epsilon, we get: \( | \int_{a}^{b} f(x) dx - r | < \frac{\epsilon}{3} \) This holds for every \(\epsilon > 0\). Hence, we can conclude that the Riemann integral of \(f\) is equal to \(r\).