Proof of Inequalities: Tips and Tricks for Checking Accuracy

[phillyolly]

 

[malicx]

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.

 

[phillyolly]

Following your recommendation, the only way I see the forward direction happening, is to write my expression in an opposite order.

 

[malicx]

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