Question

Show that every discrete space \((S, \rho)\) is complete.

Short answer

A discrete metric space is complete because Cauchy sequences stabilize and converge to a limit in the space.

Step-by-step solution

Understanding Discrete Metric

In a discrete metric space \(S, \rho\), the metric is defined such that for all points \(x, y \in S\), \ \rho(x, y) = 0 \text{ if } x = y \text{ and } \rho(x, y) = 1 \text{ if } x eq y\. This tells us that points are only zero distance apart if they are the same, and otherwise they are exactly a distance of 1 apart.

Defining Completeness of Metric Space

A metric space \(S, \rho\) is complete if every Cauchy sequence in \(S\) converges to a limit that is also in \(S\). A Cauchy sequence is a sequence where, for every positive \varepsilon\, there exists an \text{N}\ such that for all \(m, n > \text{N}\), \ \rho(x_m, x_n) < \varepsilon\.

Analyzing a Cauchy Sequence in Discrete Space

Given the metric in a discrete space, let's analyze a Cauchy sequence \(\{x_n\}\). For this sequence to be Cauchy, eventually \( \rho(x_m, x_n) < \varepsilon\) for any arbitrary small \(\varepsilon\). In the discrete metric, this means that \(\rho(x_m, x_n) = 0\) for all large \(m, n\), forcing the points to be the same, \(x_m = x_n\) for sufficiently large \(m, n\).

Showing Convergence within Discrete Space

Since \(x_m = x_n\) for all sufficiently large \(m, n\), it follows that the sequence becomes constant from some point onward. Thus, it converges to the value of these repeated elements. Since these elements are already in \(S\), the limit point is also in \(S\). Therefore, every Cauchy sequence in a discrete space \(S\) converges to a point within \(S\), showing the space is complete.

Key concepts

Cauchy sequences

In mathematics, Cauchy sequences help us understand the behavior of sequences in a metric space. A sequence is called a Cauchy sequence if the elements of the sequence get arbitrarily close to each other as the sequence progresses. You can think of this as the elements of the sequence huddling together more and more tightly, without necessarily converging to a known value immediately.
Imagine you are in a room, and people keep getting closer together as time passes, at some stage, they will form a tight group. In mathematical terms, if given any small distance (often called \( \varepsilon \)), you can eventually find a point in the sequence beyond which all elements of the sequence are closer to each other than this distance \( \varepsilon \), it qualifies as a Cauchy sequence.

In a discrete metric space, which uses the discrete metric where \( \rho(x, y) = 0 \) if \( x = y \) and \( \rho(x, y) = 1 \) if \( x eq y \), for a sequence to be Cauchy, its elements must eventually be identical. This means that from some point onward, the sequence does not change, akin to everyone standing perfectly still in our analogy.

Completeness of metric spaces

Completeness is a vital concept in the study of metric spaces. It tells us that a space doesn’t have any 'holes' concerning Cauchy sequences. Specifically, a metric space is called complete if every Cauchy sequence has a limit that is within the space itself.
Think of completeness as ensuring no matter how tightly people huddle together, they will always end up in a specific part of the room, with no falling into an invisible pit outside the floor’s boundary.

For discrete metric spaces, demonstrating completeness is straightforward. Given that any Cauchy sequence must become constant from some point, it converges within the space. This means all limit points of Cauchy sequences in a discrete space will be actual points within the space itself. Thus, discrete spaces are always complete because there's no way for a sequence’s limit to "escape" from the space.

Convergence in metric spaces

Convergence is a core aspect when dealing with sequences and metric spaces. A sequence is said to converge in a metric space if its terms get arbitrarily close to a certain point, which is then called the limit of that sequence.
A sequence starts from an initial point and, over time, gets closer and closer to a specific point in the metric space — like following a path that leads directly to a finish line.

In a discrete metric space, convergence is particularly simple. Since a sequence can only have terms at a distance of 0 or 1 from each other, a Cauchy sequence must stabilize at a single point. This means it stops wandering and fixes on a point that acts as its limit.

• Convergence to a point exists as the sequence becomes constant.
• Thus, the sequence's limit is an element of the metric space itself.

Intuitively, in a discrete space, sequences "settle down" without moving to an out-of-bounds area, ensuring their convergence within the space.