Proving a<=b when a<=b1 for all b1>b in Real Analysis

In summary, the conversation discusses how to prove that if a<=b1 for every b1 > b, then a <= b. The participants suggest using a proof by contradiction and using (a+b)/2 as b1. They ultimately come to the conclusion that a>b and b1 = (a+b)/2 leads to a contradiction.
  • #1
phygiks
16
0
Hey guys, got stuck on this question while doing homework. I would appreciate any help.
Let a,b exist in reals. Show that if a<=b1 for every b1 > b. then a <= b.

I really got nowhere. I tried letting b1(n)=b+nE where E is a infinitesimal. Then a <= b+nE for all n. Don't really know how to use any axioms here either.
 
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  • #2
Do a proof by contradiction. Suppose it's NOT true that a<=b. Then a>b. Where is (a+b)/2? Use that for b1. You certainly don't need infinitesimals.
 
  • #3
I think I got it, so a>b, let b1 = (a+b)/2, then b1 > b, but a<= (b1=(a+b)/2). Contradiction
 
  • #4
You've got it.
 

FAQ: Proving a<=b when a<=b1 for all b1>b in Real Analysis

What is Real Analysis?

Real Analysis is a branch of mathematics that deals with the study of real numbers, functions, and their properties. It is used to understand and analyze the behavior and properties of real numbers and functions in a rigorous and precise manner.

How is "a<=b when a<=b1 for all b1>b" related to Real Analysis?

This statement is a fundamental concept in Real Analysis and is known as the "least upper bound property". It states that if a set of real numbers has an upper bound, then it also has a least upper bound or supremum. This property is crucial in proving many theorems in Real Analysis.

Can you explain the logic behind proving "a<=b when a<=b1 for all b1>b"?

The proof involves using the definition of a least upper bound and the properties of real numbers. We start by assuming that a set of real numbers has an upper bound b and then show that there exists a least upper bound a. This is done by showing that a is the smallest number that is greater than or equal to all the elements in the set.

What are some common techniques used to prove "a<=b when a<=b1 for all b1>b"?

Some common techniques used in this proof include using the completeness axiom, the Archimedean property, the definition of least upper bound, and properties of real numbers such as transitivity and reflexivity.

Why is proving "a<=b when a<=b1 for all b1>b" important in Real Analysis?

This proof is important because it is the foundation for many theorems in Real Analysis. It allows us to establish the existence of supremum and infimum, which are crucial in defining and understanding the properties of real-valued functions. It also helps us in proving other theorems such as the Bolzano-Weierstrass theorem, which is used in many areas of mathematics.

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