Real Analysis Help: Proving Positive Real Numbers for Beginners

In summary, the conversation discusses how to prove that for a positive number a, there exists an integer n such that na≤x≤(n+1)a for any positive real number x. The speaker is struggling to figure out where to start and asks for help. They are given a hint to consider the set of all positive integers larger than y and use the well-ordered property of non-negative integers to prove the desired property.
  • #1
Chemistry101
15
0
1. Let a be a positive number. Prove that for each positive real number x there is an integer n such that na≤x≤(n+1)a.

I have been looking through mounds of books, but haven't figured out where to start. Our teacher just left us hanging on how to figure it out. I am severely stuck and need help on how to get the problem started.
Thanks.
 
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  • #2
What do you have to work with? Do you, for example, know that "for any y, there exist n such that [itex]n\le y\le n+1[/itex]"? If so, think about y/a.

If you don't know that, do you know that "every set of non-negative integers has a smallest member" (the "well-ordered property" of the non-negative integers). It's not too difficult to use that to prove the property I mentioned above. Think about the set of all positive integers larger than y.

Obviously, in order to prove something about positive integers, you have to use some property of positive integers!
 

1. What is Real Analysis?

Real Analysis is a branch of mathematics that deals with the study of real numbers and their properties. It involves the use of rigorous mathematical techniques and proofs to understand the nature of real numbers and their relationships with other mathematical concepts.

2. What are positive real numbers?

Positive real numbers are numbers that are greater than zero and can be expressed on the number line to the right of zero. These numbers can be either rational or irrational, but they are always greater than zero.

3. How do you prove that a number is positive real?

To prove that a number is positive real, you must show that it is greater than zero and can be expressed as a decimal or fraction. This can be done by using mathematical techniques such as induction, contrapositive, or proof by contradiction.

4. What are some common properties of positive real numbers?

Some common properties of positive real numbers include closure under addition and multiplication, the existence of an identity element (1), and the existence of inverse elements (reciprocals) for each positive real number. Additionally, positive real numbers follow the rules of commutativity, associativity, and distributivity.

5. Why is understanding positive real numbers important in real analysis?

Understanding positive real numbers is crucial in real analysis because they are the building blocks of many mathematical concepts and structures. They are also essential in solving complex problems and proving theorems in various mathematical fields, making them a fundamental aspect of higher-level mathematics.

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