Proving abs(x-y) < ε for all ε>0 in Real Analysis

In summary, the conversation involves proving the statement "abs(x-y) < ε for all ε>0, then x=y" by either considering a contradiction or using the contrapositive. The key idea is that if |x-y| > 0, choosing ε = |x-y|/2 results in a contradiction, which suggests that |x-y| must be equal to 0. This leads to the conclusion that x=y.
  • #1
mb55113
3
0
Prove that abs(x-y) < ε for all ε>0, then x=y.

I really do not know how to start this... I have tried to do the contra positive which would be If x does not equal y, then there exist a ε>0 such that abs(x-y) >= ε. Can someone help me and lead me to the right direction.
 
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  • #2
Here's how I might approach the problem. Given any two real numbers [itex]x,y \in \mathbb{R}[/itex] we clearly have that [itex]x-y \in \mathbb{R}[/itex] and similarly that [itex]|x-y| \in \mathbb{R}[/itex]. Now, suppose that [itex]|x-y| > 0[/itex], can you see how this imediately results in a contradiction? What does this contradiction suggest about the value of [itex]|x-y|[/itex]?
 
  • #3
However, if you want to prove the contrapositive, you could do it this way. Pick any two real numbers [itex]x,y[/itex] such that [itex]x-y \neq 0[/itex]. Then clearly [itex]|x-y| > 0[/itex]. Now, what happens if you choose something like [itex]\varepsilon = \frac{|x-y|}{2}[/itex]?
 
  • #4
mb55113 said:
If x does not equal y, then there exist a ε>0 such that abs(x-y) >= ε. Can someone help me and lead me to the right direction.
Quick: without thinking, tell me a positive number that is either equal to or less than 5.
 
  • #5
Ok...I tried the contrapositive...I think that i got it so suppose that abs(x-y)=m>0. Therefore we make ε=m/2 which makes abs(x-y)=m>(m/2)=ε and therefore we have found an ε>o and which makes abs(x-y)>= ε...is that right?
 

FAQ: Proving abs(x-y) < ε for all ε>0 in Real Analysis

What is real analysis?

Real analysis is a branch of mathematics that deals with the properties and behavior of real numbers and functions defined on the real numbers. It involves the study of limits, continuity, differentiation, integration, and sequences and series of real numbers.

What is the importance of proof in real analysis?

Proofs are an essential aspect of real analysis as they provide rigorous justification for mathematical statements and results. They allow us to understand the reasoning behind a theorem or concept and ensure that it is logically sound and applicable in various situations.

What are some common techniques used in proofs in real analysis?

Some common techniques used in proofs in real analysis include direct proof, proof by contradiction, mathematical induction, and proof by contrapositive. The choice of technique depends on the specific problem and the approach that best fits the situation.

How can I improve my skills in writing proofs in real analysis?

Practicing regularly and familiarizing yourself with different proof techniques is key to improving your skills in writing proofs in real analysis. It is also helpful to study and analyze well-written proofs and seek feedback from your peers or instructors.

What are some common mistakes to avoid when writing proofs in real analysis?

Some common mistakes to avoid when writing proofs in real analysis include using vague language, making assumptions without justification, and skipping steps in the proof. It is also important to carefully check for errors and ensure that the logical flow of the proof is clear.

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