Proof of Inequalities: Tips and Tricks for Checking Accuracy

In summary, the conversation revolves around a proof involving an "if and only if" statement. The first step is to address a typo in the solution and then to prove both the forward and backward directions. The speaker recommends reversing the order of the expression in the forward direction, and it is clarified that while this may work in this specific case, it is not always applicable in "if and only if" proofs. A specific example is referenced for further understanding.
  • #1
phillyolly
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  • #2
First of all, there is a typo in the second step of your solution, you lost your epsilon on the right hand side of the inequality.

Second, this is an "if and only if" statement, which means you need to prove the forward direction (Assume |x-a|<e and show a-e < x < a+e) and the backward direction (which you did). In this case the steps are all reversible, but for ''if an only if" statements in general, you need to show both directions.
 
  • #3
Following your recommendation, the only way I see the forward direction happening, is to write my expression in an opposite order.
 
  • #4
phillyolly said:
Following your recommendation, the only way I see the forward direction happening, is to write my expression in an opposite order.

Right, as I said, in this proof the steps are reversible. In general for "if and only if" proofs though, this is not the case; the forward and backward direction proofs will be different.

EDIT: Check out the proof under "Example: Division" for an illustration of what I mean: http://zimmer.csufresno.edu/~larryc/proofs/proofs.ifandonlyif.html
 

1. What is the purpose of checking a proof on inequalities?

The purpose of checking a proof on inequalities is to ensure that the mathematical argument presented is logically sound and all statements and steps are valid. This helps to avoid errors and confirm the accuracy of the proof.

2. How do I check my proof on inequalities?

To check a proof on inequalities, you should carefully read through each step of the proof and verify that it follows the rules of mathematical logic. You should also check the validity of any assumptions or definitions used in the proof.

3. What are some common mistakes to look out for when checking a proof on inequalities?

Some common mistakes to look out for when checking a proof on inequalities include incorrect application of mathematical properties, incorrect use of symbols or notation, and circular reasoning.

4. Can I use a calculator to check my proof on inequalities?

While a calculator can be useful for numerical calculations, it should not be used to check the validity of a proof on inequalities. This is because a proof relies on logical reasoning and not just numerical values.

5. What should I do if I find an error in my proof on inequalities?

If you find an error in your proof on inequalities, you should carefully review your steps and identify where the mistake was made. Then, correct the error and recheck your proof to ensure its validity.

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