Convergence, Cauchy Sequences, and Completeness

The definition of a limit for real sequences used only two things: the terms $x_n$, and the distance $|x_n-L|$ from the candidate limit. Replace $|x_n-L|$ by $d(x_n,p)$ and the definition, together with almost every proof about it, carries over to an arbitrary metric space unchanged. What is new is that some spaces have gaps a Cauchy sequence can fall into. Completeness is the property of having none, and it is the metric-space form of the least-upper-bound axiom that made $\mathbb{R}$ usable.

Sequences and their limits

The condition $d(x_n,p)<\epsilon$ says $x_n\in B(p,\epsilon )$, so convergence means: every ball about $p$, however small, contains all but finitely many terms. The number $M$ marks the tail from which the sequence stays inside the $\epsilon$-ball.

The limit is unique, and the proof is the real-line proof with $d$ in place of $|\cdot |$: if $x$ and $y$ are both limits, then for any $\epsilon$ some tail satisfies $d(x,x_n)<\epsilon /2$ and $d(x_n,y)<\epsilon /2$, so $d(x,y)\le d(x,x_n)+d(x_n,y)<\epsilon$; since $\epsilon$ was arbitrary, $d(x,y)=0$ and $x=y$. Two more facts transfer intact:

• Convergent sequences are bounded (the tail sits in one ball, the finite head in another).
• Squeeze by a null real sequence. $x_n\to p$ iff there are reals $a_n\to 0$ with $d(x_n,p)\le a_n$; this reduces metric convergence to ordinary convergence of the distances.

Componentwise convergence in ${\mathbb{R}}^n$. A sequence $x_m=(x_{m,1},\dots ,x_{m,n})$ in Euclidean space converges iff each coordinate sequence converges, in which case the limit is taken coordinate by coordinate:

One direction follows from $|y_k-x_{m,k}|\le d(y,x_m)$, the other from $d(y,x_m)<\epsilon$ once every coordinate is within $\epsilon /\sqrt{n}$. This reduces limits in ${\mathbb{R}}^n$ to $n$ limits in $\mathbb{R}$, and identifies convergence in $\mathbb{C}\cong {\mathbb{R}}^2$ with convergence of real and imaginary parts.

Convergence and the topology

Convergence can be stated with no reference to $\epsilon$ at all, using only open neighborhoods, which is why it is a topological notion.

Any open $U\ni p$ contains a ball $B(p,\epsilon )$, and the tail inside that ball is inside $U$; conversely each ball is itself an open neighborhood. The two formulations are interchangeable, and they connect convergence to closed sets and closures.

A closed set contains the limits of all its convergent sequences: if the limit $p$ lay in the open set $E^c$, some tail would be trapped in $E^c$, contradicting $\{x_n\}\subseteq E$. And the closure consists precisely of the sequential limits from $A$: given $p\in \overline{A}$, each ball $B(p,1/n)$ meets $A$ in some $x_n$, and those $x_n\to p$. This makes closed and contains its sequential limits the same statement, and recovers the closure as the reachable points.

Cauchy sequences

Convergence names a limit. Often one wants to certify that a sequence settles without naming where. The Cauchy condition does this using only the terms.

The terms bunch together: past $M$, any two of them are within $\epsilon$. Every convergent sequence is Cauchy, since if $x_n\to p$ then for $n,k\ge M$,

The converse is the question. In $\mathbb{R}$ it holds, and that was the substance of Cauchy completeness of $\mathbb{R}$. In a general metric space it can fail: the terms bunch, but the point they bunch around may not exist in the space.

Complete metric spaces

Completeness says the space has no gaps for a Cauchy sequence to fall through. The central examples are complete; the instructive failures are subspaces that omit a limit point.

For ${\mathbb{R}}^n$ the proof reduces to coordinates: a Cauchy sequence in ${\mathbb{R}}^n$ is Cauchy in each coordinate by $|x_{i,k}-x_{j,k}|\le d(x_i,x_j)$, each coordinate converges by completeness of $\mathbb{R}$, and componentwise convergence assembles the limit. For $C[a,b]$, a Cauchy sequence in the sup metric is a uniformly Cauchy sequence of functions; it converges uniformly to a limit, and the uniform limit of continuous functions is continuous, so the limit lies back in $C[a,b]$.¹

The rationals $\mathbb{Q}$ fail for the same reason: a sequence of rationals converging to $\sqrt{2}$ in $\mathbb{R}$ is Cauchy in $\mathbb{Q}$ with no rational limit, since $\sqrt{2}$ is irrational. Completeness is precisely the property that separates $\mathbb{R}$ from $\mathbb{Q}$, and the least-upper-bound property is what supplies it.

The pattern in the failures is that both omit a limit point of themselves. Adding back exactly the missing limits repairs completeness.

A Cauchy sequence in $E$ is Cauchy in $X$, so it converges in $X$; because $E$ is closed, the limit lies in $E$. This is why $(0,1]$ fails but $[0,1]$ succeeds: the latter is closed in the complete space $\mathbb{R}$.

The abstract theorem replaces an $\epsilon$-$\delta$ argument about the limit function with a single appeal to completeness.

Completion

A space with gaps can be enlarged to fill them, exactly as $\mathbb{Q}$ is enlarged to $\mathbb{R}$.

The construction mirrors the Cauchy-sequence build of $\mathbb{R}$: points of $\hat{X}$ are equivalence classes of Cauchy sequences in $X$, two sequences identified when the distance between corresponding terms tends to $0$, and $\hat{d}$ is the limit of the term-by-term distances. The original space sits inside as the classes of constant sequences, and it is dense because every Cauchy sequence is, by definition, approached by its own terms. The completion of $\mathbb{Q}$ is $\mathbb{R}$; the completion of $(0,1]$ is $[0,1]$; the completion of the polynomials on $[a,b]$ under the sup metric is all of $C[a,b]$, since by the Weierstrass approximation theorem every continuous function is a uniform limit of polynomials. Completion lets analysis assume completeness without loss of generality.

Nested sets and completeness

Completeness has a geometric consequence: nested closed balls whose radii tend to zero always contain a common point. This is the metric analogue of the nested interval property.

The centers $\{x_n\}$ form a Cauchy sequence, because for $n,k\ge M$ both lie in $C(x_M,r_M)$ and so are within $2r_M\to 0$; completeness gives a limit $p$, and $p$ lies in every $C(x_n,r_n)$ since each is closed and contains the tail. Two common points would be within every $2r_n$, forcing them equal.

This nested-ball property, completeness, and the convergence of Cauchy sequences are three views of the same fact about ${\mathbb{R}}^n$ and $C[a,b]$. Completeness together with a finiteness condition yields compactness, where every sequence has a convergent subsequence.