How do you check if a sequence is Cauchy in a metric space?

1 What is a metric space?

A metric space is a set of elements, called points, together with a function, called a metric, that assigns a non-negative number to any pair of points. The metric represents the distance between the points, and it must satisfy some basic axioms, such as symmetry, positivity, and the triangle inequality. For example, the set of real numbers with the usual absolute value function is a metric space, as well as the set of vectors with the Euclidean norm, or the set of functions with the supremum norm. Metric spaces are useful for studying various aspects of geometry, topology, analysis, and more.

2 What is a Cauchy sequence?

A Cauchy sequence is a sequence of points in a metric space that satisfies the following condition: for any positive number epsilon, there exists a natural number N such that the distance between any two terms of the sequence beyond the N-th term is less than epsilon. In other words, a Cauchy sequence is a sequence that eventually becomes as close as we want, no matter how small we make epsilon. For example, the sequence 1, 1/2, 1/3, 1/4, ... is Cauchy in the metric space of real numbers, because for any epsilon, we can choose N to be larger than 1/epsilon, and then the distance between any two terms after the N-th term will be less than 1/N, which is less than epsilon.

3 How to check if a sequence is Cauchy?

To check if a sequence is Cauchy in a metric space, we need to use the definition and the metric function. We need to show that for any epsilon > 0, we can find an N such that for any m, n > N, we have d(x_m, x_n) < epsilon, where x_m and x_n are the m-th and n-th terms of the sequence, and d is the metric function. Sometimes, this can be done by finding a general formula or inequality for d(x_m, x_n) in terms of m and n, and then solving for N in terms of epsilon. Other times, we may need to use some properties or tricks of the metric space or the sequence to simplify the calculation.

4 Examples of Cauchy sequences

Let's look at some examples of Cauchy sequences in different metric spaces. In the metric space of real numbers with the absolute value function, the sequence x_n = 1/n is Cauchy, as well as the sequence x_n = n/(n+1). In the metric space of vectors with the Euclidean norm, the sequence x_n = (1/n, 1/n^2) is Cauchy. However, in the metric space of functions with the supremum norm, the sequence x_n = sin(nx) is not Cauchy. To see this, note that d(x_m, x_n) = sup{|sin(mx) - sin(nx)| : x in [0, 2pi]}, and for any m and n that are not multiples of each other, we can find an x such that sin(mx) = 1 and sin(nx) = -1, or vice versa. Thus, d(x_m, x_n) does not depend on m and n; therefore, we cannot find an N such that for any m, n > N, we have d(x_m, x_n) < epsilon.

5 Properties of Cauchy sequences

Cauchy sequences possess some intriguing and advantageous properties that link them to other ideas in analysis. For instance, every convergent sequence is Cauchy, though the converse is not necessarily true. In a complete metric space, however, the converse is accurate. Additionally, a subsequence of a Cauchy sequence is also Cauchy and Cauchy sequences are bounded. To demonstrate this, consider that if a sequence is Cauchy, then for some N and epsilon > 0, we have d(x_m, x_n) < epsilon for any m, n > N. Therefore, we can choose M to be the maximum of d(x_1, 0), d(x_2, 0), ..., d(x_N, 0), and epsilon. Then for any n > N and n <= N, we have d(x_n, 0) < M.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?