Uniqueness of Limits of Sequences

In summary, the conversation discusses the concept of limits and how they can be used to prove that two numbers are equal. By applying the triangle inequality and choosing a small and positive number ε, it can be shown that if the difference between two limits is less than ε, then the two limits are actually the same number. This understanding helps clarify the conclusion that x' - x'' = 0.
  • #1
Fibo's Rabbit
8
0
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 homework by the way. I'm just trying to better my understanding.
 
Physics news on Phys.org
  • #2
If |x' - x''| < ε, for any small and positive number ε, then x' and x'' are the same number. IOW, x' = x''.
 
  • #3
In general, [itex]0\leq a<\epsilon[/itex] is satisfied for all [itex]\epsilon>0[/itex] implies
that [itex]a=0[/itex]. Since otherwise, if [itex]a>0[/itex] then taking [itex]\epsilon=a[/itex] makes contradiction.
 

FAQ: Uniqueness of Limits of Sequences

What is the definition of the uniqueness of limits of sequences?

The uniqueness of limits of sequences refers to the property that a sequence has only one limit, which is a single value that the terms of the sequence approach as the index increases.

How is the uniqueness of limits of sequences different from the convergence of sequences?

The uniqueness of limits of sequences is a necessary condition for convergence, but it is not sufficient. A sequence can have only one limit but still not be convergent if it does not approach that limit closely enough.

Can a sequence have more than one limit?

No, a sequence can only have one limit. This is known as the uniqueness of limits of sequences.

What happens if a sequence does not have a limit?

If a sequence does not have a limit, it is considered to be divergent. This means that the terms of the sequence do not approach a single value as the index increases.

How can the uniqueness of limits of sequences be proven?

The uniqueness of limits of sequences can be proven using the epsilon-delta definition of limits. This involves showing that for any given epsilon (a small positive number), there exists a corresponding delta (a positive number) such that the terms of the sequence will be within epsilon of the limit for all indices greater than delta.

Similar threads

B How can hyperreal numbers make infinitesimals logically sound in calculus?
Replies
24
Views
4K
B Definition of the limit of a sequence
Replies
30
Views
3K
Understanding Uniform Continuity to Formalizing Proofs
Replies
6
Views
2K
B How does the delta ε definition prove derivatives?
Replies
9
Views
3K
Does Cauchy's Theorem Confirm That Arithmetic Means of Null Sequences Are Null?
Replies
4
Views
4K
I Understanding Theorem 13 from Calculus 7th ed, R. Adams, C. Essex, 4.10
Replies
10
Views
2K
I Proof of addition of limits when the output value is infinite
Replies
8
Views
2K
Is the Limit of a Function at a Point Always Unique?
Replies
28
Views
3K
B Why do I not understand what seems to be basic Calculus?
Replies
11
Views
2K
I Spivak, Ch. 20: Understanding a step in the proof of lemma
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top