Simple proof in analysis (where did I go wrong)

In summary, the conversation discusses how to prove that if a-e < b for any e > 0, then a <= b. The approach of using proof by contradiction is suggested, but it is noted that it may not be helpful in this case. Instead, considering the cases of 0 <a-b<e and a-b <0 <e is proposed as a way to prove that a <= b.
  • #1
ych22
115
1

Homework Statement



Let a,b be any real number. Show that if a-e < b for any e > 0, then a <= b.

Homework Equations





The Attempt at a Solution



I tried a proof by contradiction, that if a>b then we violate some assumption.
Here it goes:

Suppose not.
Then a > b.
a- e > b - e
Since we assume a-e < b, thus b > b-e. But well...this seems right. Where did I go wrong?
 
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  • #2
You didn't go wrong anywhere because you didn't do anything. However, I don't think your approach would help you with the proof since if a-e > b-e and b>b-e then a-e >b. There is also the possibility that b>a-e.

I would consider to the cases
0 <a-b<e and
a-b <0 <e.

The second case is sort of trival. The first case is sort of a limit definition. So you can show a=b in the first case and a<b in the second.
 
Last edited:

1. What is a simple proof in analysis?

A simple proof in analysis is a mathematical argument that uses logical reasoning to show that a mathematical statement or theorem is true. It typically involves breaking down a complex problem into smaller, more manageable steps and providing a clear explanation of each step.

2. Why is it important to provide a simple proof in analysis?

Providing a simple proof in analysis is important because it allows others to understand and verify the validity of a mathematical statement or theorem. It also helps to identify any errors or mistakes in the reasoning or calculations.

3. Can there be more than one simple proof for a mathematical statement or theorem?

Yes, there can be multiple simple proofs for a mathematical statement or theorem. In fact, having multiple proofs can help to deepen our understanding of the mathematical concept and provide alternative approaches to solving the problem.

4. Where do most people go wrong when trying to provide a simple proof in analysis?

One common mistake when providing a simple proof in analysis is assuming that all steps are obvious and skipping over important details. It is important to clearly explain each step and justify why it is valid.

5. How can I improve my ability to provide simple proofs in analysis?

One way to improve your ability to provide simple proofs in analysis is to practice regularly and seek feedback from others. It is also helpful to study and understand different proof techniques and strategies used in analysis.

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