On the sums of elements of uncountable sets

In summary, uncountable sets are infinite sets that cannot be put into a one-to-one correspondence with counting numbers. Studying the sums of elements of uncountable sets is important for understanding their properties and has applications in mathematics. The sum of elements of an uncountable set is always infinite, and it is defined using the concept of a limit. The sum can be negative or complex depending on the elements and definition used.
  • #1
bbkrsen585
11
0
I want to prove the following proposition:

Given any uncountable set of real numbers S, there exists a countable sub-collection of numbers in S, whose sum is infinite.

Please point me in the right direction.
 
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  • #2
As a small hint, consider for each n how many elements can have absolute value smaller than 1/n
 

Related to On the sums of elements of uncountable sets

1. What are uncountable sets?

Uncountable sets are sets that have an infinite number of elements, and cannot be put into a one-to-one correspondence with the counting numbers (1, 2, 3, ...). They have a higher cardinality than countable sets, meaning they have a larger number of elements.

2. Why is it important to study the sums of elements of uncountable sets?

Studying the sums of elements of uncountable sets is important because it helps us understand the properties and behavior of these sets. It also has applications in many areas of mathematics, such as analysis, topology, and measure theory.

3. Can the sum of elements of an uncountable set be finite?

No, the sum of elements of an uncountable set is always infinite. This is because uncountable sets have an infinite number of elements, and when we add an infinite number of elements, the result will always be infinite.

4. How do we define the sum of elements of an uncountable set?

The sum of elements of an uncountable set is defined using the concept of a limit. We take a sequence of finite sums, where each sum includes more and more elements of the set, and then we take the limit of this sequence. If the limit exists, then we say that the sum of elements of the uncountable set converges, otherwise, it diverges.

5. Can the sum of elements of an uncountable set be negative or complex?

Yes, the sum of elements of an uncountable set can be negative or complex. This depends on the elements of the set and the specific definition of the sum being used. For example, in some cases, the sum may be defined using an integral, which can result in negative or complex values.

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