Here's an example that helps illustrate my question:(adsbygoogle = window.adsbygoogle || []).push({});

Prove: A sequence inRcan have at most one limit.

Proof:

Assume a sequence {x_{n}}_{n[tex]\in[/tex]N}has two limits a and b.

By definition:

-For any [tex]\epsilon[/tex]>0, there exists an N[tex]\in[/tex]Nsuch that n[tex]\geq[/tex]N implies that |x_{n}-a| < [tex]\epsilon[/tex]/2.

-A similar argument can be made for the limit b.

Thus:

|a-b| [tex]\leq[/tex] |x_{n}-a| - |x_{n}-b| < [tex]\epsilon[/tex]/2 + [tex]\epsilon[/tex]/2 = [tex]\epsilon[/tex]

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[tex]\geq[/tex]N implies that |x_{n}-a| < [tex]\epsilon[/tex], not that n[tex]\geq[/tex]N implies that |x_{n}-a| < [tex]\epsilon[/tex]/2. So is it acceptable to put [tex]\epsilon[/tex] over any number when convenient in a proof? As in the above proof, it is convenient to put [tex]\epsilon[/tex]/2 instead of just [tex]\epsilon[/tex] so that in the final step the two add up to [tex]\epsilon[/tex] 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.

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# Question Re. Simple epsilon proofs

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