Uniqueness of Limits of Sequences

[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 (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 homework by the way. I'm just trying to better my understanding.

 

[Mark44]

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

 

[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.