Problem 9 - Advanced Linear Algebra

Question

Let \(\left\\{r_{k} \mid k \in K\right\\}\) be a collection of nonnegative real numbers. If the sum on the left below converges, show that $$ \sum_{k \in K} r_{k}=\sup _{J \operatorname{fin}_{j \in K}} \sum_{k \in J} r_{k} $$

Step-by-Step Solution

Answer

The short answer to the given question is as follows: Let \(S_n = \sum_{k \in K} r_k \) for every countable subset \(n\) of \(K\), and \(T_n = \sum_{k \in J} r_k\) for every finite subset \(J\) of \(K\). Since the sum \(S_n\) converges, it implies that the sequence of partial sums, \(T_n\), also converges. We define the supremum of all finite sums of the elements in the collection as \(U = \sup_{J \operatorname{fin}_{j \in K}} T_n\), and we need to show that \(L = U\). By proving that \(L\) is the least upper bound of all \(T_n\) and \(L\) is an upper bound for all sums \(T_n\), we have shown that: \[ \sum_{k \in K} r_k =\sup _{J \operatorname{fin}_{j \in K}} \sum_{k \in J} r_k \] And the exercise is complete.

Step 1: Define variables and notations

Let's define \(S_n = \sum_{k \in K} r_k \) for every countable subset \(n\) of \(K\), and \(T_n = \sum_{k \in J} r_k\) for every finite subset \(J\) of \(K\).

Step 2: Verify convergence of the sum S_n and prove for T_n

As given, the sum \(S_n\) converges for every countable subset \(n\) of \(K\). It means that there is a finite limit \(L\), such that \(S_n\) approaches to \(L\) as \(n\) increases. Thus, we have: \[ \lim_{n\to\infty} S_n = L \] Since the series \(S_n\) converges, it implies that the sequence of partial sums, \(T_n\), also converges. Therefore, we can say that for every \(T_n\), \(T_n\) is an upper bound for the set of finite sums of the elements in the collection.

Step 3: Show the supremum of T_n

Since every finite subset of \(K\) is indexed by \(J\) and has a corresponding partial sum \(T_n\), the supremum of all the finite sums can be denoted as \(U = \sup_{J \operatorname{fin}_{j \in K}} T_n\). We now need to show that \(L = U\), which would imply that the sum on the left converges and is equal to the supremum of the sums of all finite subsets of the collection.

Step 4: Prove that L is an upper bound of T_n

Let's first show that \(L\) is an upper bound for all sums \(T_n\). We know that as \(n \rightarrow \infty\), \(S_n\) approaches \(L\). Since \(T_n\) is a partial sum, it means that, for any finite subset J, \[ T_n \leq S_n \] This implies that: \[ T_n \leq L \] for every finite subset J. This shows that \(L\) is an upper bound for all finite sums \(T_n\).

Step 5: Prove that L is the least upper bound of T_n

Now, we need to show that \(L\) is the least upper bound of \(T_n\). By definition of supremum, it is sufficient to prove that for any \(\epsilon > 0\), there exists a finite sum \(T_m\) such that \(L - \epsilon < T_m\). Since \(S_n\) converges to \(L\), we know that there exists an \(n\) such that: \[ L - \epsilon < S_n \] and because \(T_m \leq S_n\), we can find a finite sum \(T_m\) such that: \[ L - \epsilon < T_m \] This implies that \(L\) is the least upper bound of \(T_n\).

Step 6: Conclusion

By proving that \(L\) is the least upper bound of all \(T_n\) and \(L\) is an upper bound for all sums \(T_n\), we have shown that: \[ \sum_{k \in K} r_k =\sup _{J \operatorname{fin}_{j \in K}} \sum_{k \in J} r_k \] And the exercise is complete.

Key Concepts

partial sum

In the world of sequences and series, a **partial sum** is an essential concept. It refers to the sum of the first few terms of a series. For instance, if you have a series given as \(a_1 + a_2 + a_3 + \ldots\ + a_n\), the partial sum \(S_n\) would be the sum of the first \(n\) terms: \(S_n = a_1 + a_2 + a_3 + \ldots\ + a_n\).

Partial sums help in understanding the behavior of a series as they offer a snapshot of the sum up to a certain point. By analyzing partial sums, we can determine if a series converges or diverges.

In the context of convergent series, the behavior of the partial sums provides significant insights. If the sequence of partial sums approaches a specific number as \(n\) tends to infinity, the series is said to converge. Otherwise, it diverges. This concept of partial sums is crucial when proving properties like the least upper bound in real analysis.

supremum

The **supremum** of a set is often explained as the *least upper bound*. It is the smallest number that is greater than or equal to every element in a set. Unlike the maximum, which must be an element of the set, the supremum can exist outside the set.

For example, consider the set of numbers \( \{ x \mid x < 1 \} \). The supremum is 1, because 1 is the smallest number that is not less than any number in the set. However, 1 is not contained in the set, showing how the supremum can be distinct from the maximum.

In the given exercise, finding the supremum involves determining the least upper bound of the sums \(T_n\), where each \(T_n\) represents the sum over a finite subset of real numbers. By showcasing that \(L\) is the least upper bound of these finite sums, we connect the concept of supremum with the convergence of the series.

least upper bound

The notion of **least upper bound** is closely related to the concept of supremum. It signifies the smallest value that bounds a set of numbers from above. This value does not need to be part of the set.

This idea plays a significant role in real analysis, especially when dealing with infinite sets or sums. To find the least upper bound, one must ensure there is no smaller number that can act as an upper bound for the set.

In the context of the exercise, after proving that \(L\) is an upper bound for all finite sums \(T_n\), we further demonstrate that \(L\) is also the least of all possible upper bounds. This is shown using a typical \(ε\)-proof, which involves demonstrating that for any value arbitrarily close to \(L\), a sum exists that is slightly lesser than \(L\), but greater than any possible alternative bound. This solidifies \(L\) as the least upper bound of the series.

real analysis

**Real analysis** is a branch of mathematics dealing with real numbers and the sequences and series that connect numbers. It explores deeper results beyond simple arithmetic, focusing on functions, series, and sequences within the real number system.

Key topics in real analysis include:

• Convergence and divergence of series
• Limits and continuity
• Integration and differentiation
• Supremum and infimum (least upper bound and greatest lower bound)

Real analysis provides the toolset for rigorously tackling problems involving infinite sums like the one in the exercise. It allows mathematicians to prove convergence, explore bounds, and handle infinite series comprehensively and logically.

The exercise you encountered required concepts of real analysis such as determining convergence and understanding the supremum. These tools ensure the correctness and completeness of numerical series handling, ensuring a robust foundation for mathematical reasoning.