91Ó°ÊÓ

In the realm of metric spaces, understanding the closure of a set is paramount. The closure of a set, denoted as \( \bar{S} \), refers to the smallest closed set that contains the original set \( S \). A closed set in this context is one that includes all its limit points—points where sequences in \( S \) converge to.
To visualize this, imagine \( S \) as a swarm of points scattered within a boundary. The closure encompasses all these points plus any outer edge or limit points reached by sequences originating from \( S \). This ensures that \( \bar{S} \) is robust, capturing every possible pathway these sequences can take.
For example, in real numbers, the closure of an open interval \((a, b) \) is the closed interval \([a, b]\), which includes all the values between \(a\) and \(b\) as well as the endpoints themselves, securing no gaps at the boundaries.

A core component of a set's closure, limit points add depth to understanding sequence behaviors within metric spaces. A limit point of a set \( S \) is a point that can be approached arbitrarily closely by elements within \( S \) itself. This means for any chosen proximity, or epsilon, there exist points in \( S \) within that distance from the limit point.

• For instance, consider the set of rational numbers within the interval \((0, 1)\). The irrational numbers, although not inside \( S \), are limit points, as subsets of the rationals can get infinitely close to any irrational number.
• Another example lies in sequences. If a sequence of points within a set converges to a particular point, that point serves as a limit point of the set.

Limit points ensure the completeness of a closure, preventing any possible 'leaks' where sequences could escape outside the boundary drawn by \( S \). In practical terms, they safeguard the defined boundaries of a set in metric spaces.

The triangle inequality is a fundamental property in metric spaces, pivotal for proving relationships like diameters in this exercise. This principle states that for any three points \( x, y, \) and \( z \) in a metric space, the direct distance between two points is always less than or equal to the distance traveled via an intermediary point. Formally, it is expressed as:
\[d(x, y) \leq d(x, z) + d(z, y)\]This rule resembles the geometric property seen in triangles: any side length must be less than or equal to the sum of the other two sides.

• In the context of this problem, when \( x \) and \( y \) are points within the closure \( \bar{S} \), sequences \( x_n \) and \( y_n \) drawn from \( S \) converge to \( x \) and \( y \). Here, the triangle inequality helps bridge the direct distance between these converging sequences and their limit points.
• This makes it a cornerstone in affirming the equality between the diameter of \( S \) and its closure.

The triangle inequality thus seamlessly integrates into metric space analyses, providing a logical scaffolding for complex set relationships like the ones explored here.