Proving InfE <= SupE, Proving 1/n < a < n, and Proving N = E

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In summary, the conversation discusses three proofs. The first proof is to show that the infimum of a nonempty subset E of real numbers is less than or equal to the supremum of E. The second proof is to show that for any positive number a, there exists a natural number n such that 1/n is less than a and n is greater than a. The third proof is to show that the set of natural numbers, denoted by N, is equal to the set E = {1,2,3,4...} which is an inductive set. The principle of mathematical induction states that if M is any subset of N that is inductive, then M = N. However, the proof shows that N =
  • #1
mollysmiith
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1. Let E be a nonempty subset of R (real numbers)

Prove that infE <= supE

2. Prove that if a > 0 then there exists n element N (natural) such that 1/n < a < n

3. A subset E of te real numbers R is an inductive set if

i) 1 element E
ii) If x element E then x + 1 element E

A real number is called a natural number if it belongs to every inductive set. The set of natural numbers is denoted by N. Recall that the principle of mathematical inductions says that if M is any subset of N that is an inductive set then M = N. Show that N = E, where E = {1,2,3,4...}

Any help would be greatly appreciated ! :)
 
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  • #2
Please show us your attempt at a solution.
 
  • #3
i can not some up with anything i am not sure where to start for any of them
 
  • #4
Do you know the definitions of inf A and sup A ?
 

1. What does it mean to prove InfE <= SupE?

Proving InfE <= SupE means to show that the infimum (greatest lower bound) of a set of numbers is less than or equal to the supremum (least upper bound) of the same set of numbers. In other words, it means to prove that the minimum value in a set is less than or equal to the maximum value in that same set.

2. How do you prove 1/n < a < n?

To prove 1/n < a < n, you must show that for any positive integer n, there exists a real number a such that 1/n is less than a and a is less than n. This can be done by finding a specific value for a that satisfies the inequality, or by using algebraic manipulations to show that the inequality holds for all possible values of n and a.

3. What does it mean to prove N = E?

Proving N = E means to show that two sets, N and E, are equal. This means that every element in set N is also in set E, and vice versa. In other words, the two sets contain the exact same elements.

4. What is the significance of proving InfE <= SupE, 1/n < a < n, and N = E?

Proving these statements is important in many areas of mathematics, including analysis, calculus, and set theory. It allows us to make precise statements about the relationships between sets of numbers and to establish important properties of these sets. In particular, these proofs are often used to show that a given set is bounded, or that it contains a minimum or maximum element.

5. Can you provide an example of a proof for InfE <= SupE, 1/n < a < n, or N = E?

Yes, for example, to prove that InfE <= SupE, we can show that the infimum of a set of numbers is always less than or equal to the supremum of that same set. This can be done by considering the definition of infimum and supremum, and using logical reasoning to show that the infimum is always less than or equal to the supremum.

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