Demonstrating Metrics and Completeness in Real Analysis

Understanding the concept of completeness in metric spaces is fundamental to real analysis. In this article, we explore how to determine the completeness of metric spaces by examining specific examples. We will use the interval -1 and explore three different metrics, demonstrating their properties and verifying their completeness.

Step 1: Showing the Metric dx,y |x - y| is Not Complete on -1

We begin by identifying a sequence that demonstrates the lack of completeness of the metric dx,y.

Step 1.1: Sequence Definition

Consider the sequence defined by:

$x_n 1 - frac{1}{n} quad text{for } n 1, 2, 3, ldots$

This sequence lies entirely within -1

Step 1.2: Sequence is Cauchy

We show that xn is a Cauchy sequence with respect to the metric d:

$dx_n x_m |x_n - x_m| left| left(1 - frac{1}{n}right) - left(1 - frac{1}{m}right) right| left| frac{1}{m} - frac{1}{n} right| to 0 quad text{as } n, m to infty$

This confirms that xn is a Cauchy sequence in -1 .

Step 1.3: Limit Out of Interval

The limit of xn is:

$lim_{n to infty} x_n 1$

Since 1 is not in the interval -1 , this shows that the metric dx,y |x - y| is not complete on -1 .

Step 2: Showing the Metric Dx,y tanfrac{pi x}{2} - tanfrac{pi y}{2} is Complete

To demonstrate that the metric Dx,y is complete, we first establish bijective mapping properties of the function tanfrac{pi x}{2} and explicitly show how this metric behaves.

Step 2.1: Bijective Mapping

The function tanfrac{pi x}{2} maps -1 bijectively to -infty . This means that any Cauchy sequence in -1 under the metric Dx,y can be mapped to a Cauchy sequence in -infty .

Step 2.2: Convergence of Cauchy Sequence

Suppose xn is a Cauchy sequence with respect to D. For any ε > 0, there exists an N such that for all n, m ≥ N, $Dx_n x_m This implies that tanfrac{pi x_n}{2} - tanfrac{pi x_m}{2} . Since the tangent function is continuous, the sequence tanfrac{pi x_n}{2} converges to some limit L ≥ -infty.

Step 2.3: Inverse Mapping and Convergence

By the continuity of the inverse function (arctangent), there exists a unique x ∈ -1 such that tanfrac{pi x}{2} L. Thus, the original sequence xn converges to x ∈ -1 , showing that the metric Dx,y is complete.

Step 3: Showing the Metric Dx,y x - y frac{1}{1-x} - frac{1}{1-y} is Equivalent and Complete

Finally, we show that this metric is equivalent to the Euclidean metric and complete.

Step 3.1: Equivalence

To show that the metric Dx,y is equivalent to dx,y |x - y|, we need to demonstrate that there exist constants c1, c2 ≥ 0 such that for all x, y ∈ -1 : $c1 |x - y| ≤ D(x,y) ≤ c2 |x - y|$

The term |x - y| is straightforward. For the other terms frac{1}{1-x} - frac{1}{1-y}, we can bound them using the Mean Value Theorem, showing they do not grow faster than |x - y| as x, y approach the endpoints of the interval.

Step 3.2: Completeness

Similar to the previous metric, if xn is Cauchy with respect to this metric, the terms involving |x - y| ensure that xn converges to some limit in -1 .

Dx,y tanfrac{pi x}{2} - tanfrac{pi y}{2} and Dx,y x - y frac{1}{1-x} - frac{1}{1-y} are both equivalent and complete on -1 .