How do you prove the limit of a sequence is unique?

How do you prove the limit of a sequence is unique?

Theorem 3.1 If a sequence of real numbers {an}n∈N has a limit, then this limit is unique. Proof by contradiction. We hope to prove “For all convergent sequences the limit is unique”. The negation of this is “There exists at least one convergent sequence which does not have a unique limit”.

How do you show a sequence converges in a metric space?

A sequence {xn} in a metric space (X,d) is said to converge to a point p∈X, if for every ϵ>0, there exists an M∈N such that d(xn,p)<ϵ for all n≥M. The point p is said to be the limit of {xn}. We write limn→∞xn:=p.

How do you show a metric space is not complete?

A metric space (X, ϱ) is said to be complete if every Cauchy sequence (xn) in (X, ϱ) converges to a limit α ∈ X. There are incomplete metric spaces. If a metric space (X, ϱ) is not complete then it has Cauchy sequences that do not converge. This means, in a sense, that there are gaps (or missing elements) in X.

What makes a limit unique?

2. Theorem on uniqueness of limits. The theorem on the uniqueness of limits says that a sequence ( ) can have at most one limit. As a consequence, to show that a sequence ( ) does not converge to some number (such as 12 say) it suffices to show that does converge to a different number such as 13 .

What does a unique limit mean?

The uniqueness theorem for limits states that if the limit of exists at (in the sense of existence as a finite real number) then it is unique.

What is convergence in metric space?

Definition. A sequence (xn) of points in a metric space (X, d) converges to a limit α if the real sequence (|d(xn, α)| converges to 0 in R. Remark. If you insist on “back to basics” this reads: Given ε > 0 there exists N ∈ N such that if n > N then we have d(xn, α) < ε.

What is a sequence in metric space?

A sequence in a metric space X is a function x:N→X. In the usual notation for functions the value of the function x at the integer n is written x(n), but whe we discuss sequences we will always write xn instead of x(n).

How do you complete metric spaces?

A metric space (X, d) is complete if any of the following equivalent conditions are satisfied:

1. Every Cauchy sequence of points in X has a limit that is also in X.
2. Every Cauchy sequence in X converges in X (that is, to some point of X).
3. The expansion constant of (X, d) is ≤ 2.

Are complete metric spaces closed?

A metric space is complete if and only if it is closed in every space containing it. Indeed, suppose X⊂Y and X is complete. Take any sequence in X that has a limit in Y. Since it converges, it is a Cauchy sequence, hence it has a limit in X.

Are sequential limits unique in the metric space?

Proposition In a metric space, sequential limits are unique. Proposition That a sequence {xn} converges in a metric space (X,d)to a point x0 is equivalent to the condition that for each > 0 there is a natural number N such that N n implies d(xn, x0) < . Examples In either the reals or complexes if |r| < 1, then rn 0.

How do you find the limit of xn of a sequence?

The sequence yk consists of all even entries, and, since yk = 1 for all k ∈ N, we have yk → 1. If the sequence xn converges to some number ℓ, then this number must be ℓ = + 1. On the other hand the subsequence zk = x2k − 1 = − 1 converges to − 1, so any limit of the sequence xn must be ℓ = − 1.

How do you know if a sequence is bounded?

Since a sequence in a metric space (X;d) is a function from N into X, the de nition of a bounded function that we’ve just given yields the result that a sequence fx. ngin a metric space (X;d) is bounded if and only if 9x 2X;M2R : 8n2N : d(x.

What is the set of values of the sequence xn?

It is important to distinguish this set from the sequence itself. For example, if X = R, and xn = 1 for all n ∈ N, then the sequence xn is 1, 1, 1, … , i.e. an infinite sequence of ones. The set of values of this sequence is {1}, which is a subset of R with only one element.