Question Re. Simple epsilon proofs

[Newtime]

Here's an example that helps illustrate my question:

Prove: A sequence in R can have at most one limit.

Proof:

Assume a sequence {x_n}_n\inN has two limits a and b.
By definition:

-For any \epsilon>0, there exists an N\inN such that n\geqN implies that |x_n-a| < \epsilon/2.
-A similar argument can be made for the limit b.

Thus:

|a-b| \leq |x_n-a| - |x_n-b| < \epsilon/2 + \epsilon/2 = \epsilon

Thus a=b.


Now here's my question...in the step immediately following "By definition," the actual definition of a limit of a sequence shows that n\geqN implies that |x_n-a| < \epsilon, not that n\geqN implies that |x_n-a| < \epsilon/2. So is it acceptable to put \epsilon over any number when convenient in a proof? As in the above proof, it is convenient to put \epsilon/2 instead of just \epsilon so that in the final step the two add up to \epsilon and show that a=b.

Thanks.

edit: sorry for the awkward formatting, if anything is unclear, let me know I'll explain it in words.

 

[snipez90]

Yes, if the limit exists, then you can replace epsilon with any positive number (so epsilon/2 is of course positive), because that's simply what epsilon typically denotes. Many people want the last step of a proof to be just epsilon, but often times it doesn't matter. In this case, it doesn't matter whether we have epsilon or 2*epsilon at the very end, since epsilon is arbitrary anyways. Note you have a typo in the leftmost inequality at the final step of the proof.

 

[Newtime]

Yes, if the limit exists, then you can replace epsilon with any positive number (so epsilon/2 is of course positive), because that's simply what epsilon typically denotes. Many people want the last step of a proof to be just epsilon, but often times it doesn't matter. In this case, it doesn't matter whether we have epsilon or 2*epsilon at the very end, since epsilon is arbitrary anyways. Note you have a typo in the leftmost inequality at the final step of the proof.

Thanks for the reply, that's what I thought but wanted to hear it from someone who actually knew before I went on doing proofs on an assumption.