Problem 16 - Basic Analysis: Introduction to Real Analysis

Question

Prove Proposition 7.4.6. That is, let \((X, d)\) be a complete metric space and \(E \subset X\) a closed set. Show that \(E\) with the subspace metric is a complete metric space.

Step-by-Step Solution

Answer

Closed subsets of complete metric spaces are complete with the subspace metric.

Step 1: Understand Definitions

Recall the definitions: A metric space \((X, d)\) is complete if every Cauchy sequence in \(X\) converges to a limit that is also within \(X\). A subset \(E\) of a metric space \((X, d)\) is closed if every limit of a convergent sequence of points in \(E\) is also in \(E\). In the subspace metric, the metric \(d\) is restricted to \(E\).

Step 2: Select a Cauchy Sequence in E

Select a sequence \((x_n)\) that is a Cauchy sequence in \(E\) with the subspace metric. We aim to show that this sequence converges to a point in \(E\).

Step 3: Sequence in X

Since \(E\subset X\) and \((x_n)\) is a sequence in \(E\), it is also a Cauchy sequence in \(X\) because \(d_E\) is a restriction of \(d\). Therefore, \((x_n)\) is a Cauchy sequence in \(X\).

Step 4: Use Completeness of X

Since \((X, d)\) is a complete metric space, the Cauchy sequence \((x_n)\) converges to some limit \(x_0 \in X\).

Step 5: Show Limit is in E

Since \((x_n)\) is entirely contained in the closed set \(E\), and \(E\) is closed in \(X\), the limit \(x_0\) of \((x_n)\) must also be in \(E\) by the definition of a closed set.

Step 6: Conclude Completeness of E

Thus, every Cauchy sequence in \(E\) converges to a limit in \(E\), meaning \((E, d_E)\) is a complete metric space where \(d_E\) is the subspace metric of \(d\).

Key Concepts

Cauchy Sequence

A Cauchy sequence is a fundamental concept in analysis. It is a sequence of points in a metric space that eventually gets arbitrarily close to each other. In more formal terms, a sequence \(x_n\) is a Cauchy sequence if for every positive number \varepsilon\, there exists an integer \N\ such that for all integers \m, n \geq N\, the distance \(d(x_m, x_n) < \varepsilon\) holds true.

• This concept is essential because it helps us understand convergence, even if we do not know what the sequence converges to.
• Cauchy sequences are crucial when discussing the completeness of metric spaces, as demonstrated in the original exercise.

In a complete metric space, any Cauchy sequence is guaranteed to converge to a limit that is part of the space. This is why checking if a sequence is Cauchy can often be a precursor to establishing its convergence.

Closed Set

In the world of topology and metric spaces, a closed set is an important concept. A set \(E\) within a metric space is called closed if it contains all its limit points. This means that if a sequence of points in \(E\) is convergent, then the limit of that sequence is also in \(E\).

• Closed sets are always complete when they are considered with the subspace metric.
• Being able to identify closed sets can simplify the process of showing completeness for subspaces.

Consider how closed sets are used in proving the proposition mentioned in the original exercise. The property that the limit of any convergent sequence in a closed set stays within that set helps show that a subspace is complete when equipped with the subspace metric.

Convergent Sequence

A convergent sequence in a metric space is a sequence that approaches a particular point, called the limit, as the sequence progresses. Formally, a sequence \(x_n\) in a metric space \(X\) converges to a limit \(L\) if for every positive number \varepsilon\, there exists an integer \N\ such that for all integers \ greater than or equal to \N\, the distance \(d(x_n, L) < \varepsilon\).

• The limit \(L\) is in the same metric space as the sequence.
• Convergent sequences in complete metric spaces never "escape" the space, due to completeness.

Understanding convergent sequences is vital because they help assure us that certain sequences with desirable properties (like being Cauchy) do indeed shrink down to defined points, rather than stretching out indefinitely. This plays a key role in proofs that show completeness of a subspace, as noted in the exercise solution.

Subspace Metric

When dealing with subsets of a metric space, we often use what is known as a subspace metric. This is essentially the same distance function as the original space, but now it is restricted to a subset \(E\). If \(d\) is the metric on the space \(X\), then for any two points \(x, y \in E\), the subspace metric \(d_E(x, y)\) is the same as \(d(x, y)\).

• Using a subspace metric allows us to examine properties of \(E\) with the same underlying structure as \(X\).
• The subspace metric retains all distance comparisons within \(E\), aiding in analyses like establishing completeness.

This notion helps facilitate proofs such as showing a subset is complete when originally part of a complete metric space, as outlined in the textbook solution. It becomes particularly handy in conjunction with closed sets, since if \(E\) is closed and you use the subspace metric, the subspace inherits completeness from the larger space.