...Then the sequence {*„} is a Cauchy sequence in X that does not converge in X. A metric space (X, d) is said to be complete if every Cauchy sequence in X converges in X. The simple examples of complete metric spaces are : (a) R or С with the usual metric d (x, y)...

...sequence if for every e > 0 there is an N(e) such that ||un - uj| < e for every m, n > N The space X is said to be complete if every Cauchy sequence in X converges, that is, has a limit that is an element, say u, of X. In this case ||un - u|| — » 0 and we will...

...xm-xn <e, That is, {xn } is a Cauchy sequence in X if and only if II xm -xn || -»0 as (v) The space X is said to be complete if every Cauchy sequence in X converges to an element in A". We deduce, from the linear space structure on A", that: (/) Given x0 e X and r...

...Cauchy sequence has a convergent subsequence, then the sequence itself is convergent. A metric space A' is said to be complete if every Cauchy sequence in X converges in X. Loosely speaking, we can say that a metric space X is complete if every sequence in X which tries...

...characterized similarly.) {Hint: Theorem 6.1.23 and §6.1. Exercise 3.} 6.3. Complete Metric Spaces Recall that a metric space is said to be complete if every Cauchy sequence in the space is convergent (6.1.25). Here is another characterization of completeness: 6.3.1. Theorem....

...Cauchy sequence if for each e > 0 there exists an integer N such that d(xn'xm) < e whenever n,m > N. A metric space is said to be complete if every Cauchy sequence converges to some element in the space. Theorem 1.1.1 (Baire) If M is a complete metric space, the...

...§2. The notion of Cauchy sequence can be defined just as we did in the text, and a metric space X is said to be complete if every Cauchy sequence in X converges. We denote the distance between two points by d(xi,X2). Exercise VI. 4. 5 (The Semiparallelogram Law)...

...between the points x and v. A metric space is a pair (X, d), where X is a set and d is a metric on X. A metric space is said to be complete if every Cauchy sequence in it converges. If Y is a subset of X, then the restriction of d to Y x Y is called the induced metric...

...analogous to that of R. First we define the completeness of En . DEFINITION 13.12. The metric space X is said to be complete if every Cauchy sequence in X converges to a point in X. In the terminology of Definition 13.12, En is a complete metric space by Theorem 13.9....

...equivalent to the requirement that lim d(xn,xm) = 0. n,m—xx> (b) The space X (or the associated metric d) is said to be "complete", if every Cauchy sequence in X converges to a point in the space. REMARK 1.4.15 Clearly every Cauchy sequence is bounded, ie i sup d(xn,xm)...