Cauchy sequence

If this is a Cauchy sequence for the m-adic topology on A A2 then the limit I will... 

Let x S be a Cauchy sequence then the image sequence n) in ft is Cauchy, hence has a limit which we may assume to be zero, after possibly subtracting a constant sequence from fn. We have fn ( m [Pg.159]

If there is a metric, we may define convergence. Given a sequence of elements x, e S, is there an element towards which the series converges It turns out that a crucial question is this assume that we know that the sequence is a so-called Cauchy sequence, i.e ... 

If a Cauchy sequence converges, using the distance defined by x -xm, in a scalar product space, then it is a Hilbert space. 

Thus, we conclude that the sequence of elements m , generated by the minimal residual method, is a Cauchy sequence, because the distance between any two elements goes to zero, m — m —> 0, as f,n oo (see Appendix A, section A.2). Since the Hilbert space M is a complete linear space, the Cauchy sequence m converges to the element m G M m — in, if n —+ oo. 

A.2.2 Convergence, Cauchy sequences and completeness We begin with several definitions. 

Definition 32 Any sequence in which the distance between any two elements tends to zero, — 0, as k,j —> oo, is called a Cauchy sequence. 

It can be proved that any convergent sequence is a Cauchy sequence. In fact, from the triangle inequality, we can write... 

On the other hand, there can exist Cauchy sequences of elements which do not converge to an element in the metric space. For example, let us consider as a metric space the internal part of the geometric 3-D ball B without a boundary. We can introduce series of points Si, 2, S3,... which converge to the element Sq located at the boundary. Obviously, the set 81,82,83,... forms a Cauchy sequence, but it converges to the element sq outside our metric space B. From this point of view we can call B an incomplete metric space. 

Definition 33 A metric space is said to be incomplete if there are Cauchy sequences in it that do not converge to an element of this metric space. Conversely, a space M is complete if every Cauchy sequence converges to an element of the space. 

This means that every Cauchy sequence in a Banach space converges to an element of this space. 

The linear nonned space equipped with the inner product is called a pre-Hilbert space. In order to obtain a Hilbert space we require that the space L be complete, in other words, every Cauchy sequence of elements from L must converge to an element of this space. So we arrive at the following definition. 

Since r < 1 it is evident from the last formula, that x is a Cauchy sequence, and by the completeness of X, there exists a point x in X such that x x. 

A normed space in which every Cauchy sequence converges (to a point in the space) is said to be complete and termed a Banach space. 

Let us imagine an infinite sequence of functions (i.e., vectors) fl, fl, /s, in a unitary space (Fig. B.l). The sequence will be called a Cauchy sequence if for a given e > 0, a natural number N can be found, such that for i > N, we will have /,+i - < e. In other words, in a Cauchy sequence, the distances between the consecutive vectors (functions) decrease, when we go to sufficiently large indices i.e., the functions become more and more similar to each other. If the converging Cauchy sequences have their limits... 

One of the pioneers in the fuzzy distance approach that preserves uncertainty was Voxman (1998) which addresses principles of distance on the fuzzy point of view and treats the principle of convergence on the view of the Cauchy sequences. He was the first to propwse a fuzzy distance between fuzzy numbers. 

An operation to add all the limits of Cauchy sequences to X, d) is referred to as a completion. We recall that the space of real numbers R is a completion of the space of rational numbers Q under the uniform norm . ... 

Therefore, for a fixed j, ) forms a Cauchy sequence. Since the spaces... 

---