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Description of convergent sequences and Cauchy sequences in metric spaces. Description of complete spaces.

Convergence

The concept of convergence evaluates whether a sequence of elements is getting “closer” to a given point or not.

Definition 1 Assume a metric space ( X , d ) and a countably infinite sequence of elements { x n } : = { x n , n = 1 , 2 , 3 , ... } X . The sequence { x n } is said to converge to x X if for any ϵ > 0 there exists an integer n 0 Z + such that d ( x , x n ) < ϵ for all n n 0 . A convergent sequence can be denoted as lim n x n = x or x n n x .

Illustration of a convergent sequence { x i } .

Note that in the definition n 0 is implicitly dependent on ϵ , and therefore is sometimes written as n 0 ( ϵ ) . Note also that the convergence of a sequence depends on both the space X and the metric d : a sequence that is convergent in one space may not be convergent in another, and a sequence that is convergent under some metric may not be convergent under another. Finally, one can abbreviate the notation of convergence to x n x when the index variable n is obvious.

Example 1 In the metric space ( R , d 0 ) where d 0 ( x , y ) = | x - y | , the sequence x n = 1 / n gives x n n 0 : fix ϵ and let n 0 > 1 / ϵ (i.e., the smallest integer that is larger than 1 / ϵ ). If n n 0 then

d 0 ( 0 , x n ) = | 0 - x n | = | x n | = x n = 1 n 1 n 0 < 1 1 / ϵ 1 1 / ϵ = ϵ ,

verifying the definition. So by setting n 0 ( ϵ ) > 1 / ϵ , we have shown that { x n } is a convergent sequence.

Example 2 Here are some examples of non-convergent sequences in ( R , d 0 ) :

  • x n = n 2 diverges as n , as it constantly increases.
  • x n = 1 + ( - 1 ) n (i.e., the sequence { x n } = { 0 , 2 , 0 , 2 , ... } ) diverges since for ϵ < 1 there does not exist an n 0 that holds the definition for any choice of limit x . More explicitly, assume that a limit x exists. If x [ 0 , 2 ] then for any ϵ 2 one sees that for either even or odd values of n we have d ( x , x n ) > ϵ , and so no n 0 holds the definition. If x [ 0 , 2 ] then select ϵ = 1 2 min ( x , 2 - x ) . We will have that d ( x , x n ) ϵ for all n , and so no n 0 can hold the definition. Thus, the sequence does not converge.

Theorem 1 If a sequence converges, then its limit is unique.

Proof: Assume for the sake of contradiction that x n x and x n y , with x y . Pick an arbitrary ϵ > 0 , and so for the two limits we must be able to find n 0 and n 0 ' , respectively, such that d ( x , x n ) < ϵ / 2 if n > n 0 and d ( y , x n ) < ϵ / 2 if n > n 0 ' . Pick n * > max ( n 0 , n 0 ' ) ; using the triangle inequality, we get that d ( x , y ) d ( x , x n * ) + d ( x n * , y ) < ϵ / 2 + ϵ / 2 = ϵ . Since for each ϵ we can find such an n * , it follows that d ( x , y ) < ϵ for all ϵ > 0 . Thus, we must have d ( x , y ) = 0 and x = y , and so the two limits are the same and the limit must be unique.

Cauchy sequences

The concept of a Cauchy sequence is more subtle than a convergent sequence: each pair of consecutive elements must have a distance smaller than or equal than that of any previous pair.

Definition 2 A sequence { x n } is a Cauchy sequence if for any ϵ > 0 there exists an n 0 Z + such that for all j , k n 0 we have d ( x j , x k ) < ϵ .

As before, the choice of n 0 depends on ϵ , and whether a sequence is Cauchy depends on the metric space ( X , d ) . That being said, there is a connection between Cauchy sequences and convergent sequences.

Theorem 2 Every convergent sequence is a Cauchy sequence.

Questions & Answers

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Source:  OpenStax, Signal theory. OpenStax CNX. Oct 18, 2013 Download for free at http://legacy.cnx.org/content/col11542/1.3
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