Uniqueness of Limits of Sequences

1. Feb 13, 2012

Fibo's Rabbit

Here is the proof provided in my textbook that I don't really understand.

Suppose that x' and x'' are both limits of (xn). For each ε > 0 there must exist K' such that | xn - x' | < ε/2 for all n ≥ K', and there exists K'' such that | xn - 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' - xn + xn - x'' | ≤ | x' - xn | + | xn - 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.

2. Feb 13, 2012

Staff: Mentor

If |x' - x''| < ε, for any small and positive number ε, then x' and x'' are the same number. IOW, x' = x''.

3. Feb 14, 2012

LikeMath

In general, $0\leq a<\epsilon$ is satisfied for all $\epsilon>0$ implies
that $a=0$. Since otherwise, if $a>0$ then taking $\epsilon=a$ makes contradiction.