Here is the proof provided in my textbook that I don't really understand.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose that x' and x'' are both limits of (x_{n}). For each ε > 0 there must exist K' such that | x_{n}- x' | < ε/2 for all n ≥ K', and there exists K'' such that | x_{n}- x'' | < ε/2 for all n ≥ K''. We let K be the larger of K' and K''. Then for n ≥ K we apply the triangle inequality to get:

| x' - x'' | = |x' - x_{n}+ x_{n}- x'' | ≤ | x' - x_{n}| + | x_{n}- x'' | < ε/2 + ε/2 = ε

Since ε > 0 is an arbitrary positive number we conclude that x' - x'' = 0.Q.E.D.

***I understand how they got to the conclusion |x' - x''| < ε. What I don't understand is how they can conclude from that that x' -x'' = 0.

Any help is much appreciated. This isn't hw by the way. I'm just trying to better my understanding.

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# Uniqueness of Limits of Sequences

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