The Real Line as a Metric Space

From Absolute Differences to a Metric Space

In earlier tutorials, distance between real numbers appeared through expressions such as \(|a_n-L|\) in the definition of sequence convergence. The real line can be viewed more systematically as a set equipped with a distance function. This viewpoint is useful because it separates the idea of distance from the particular formula for distance, while preserving the estimates that make analysis work.

Here we formalize that idea. We will verify that the familiar absolute-value distance satisfies the defining axioms of a metric, then see how its open balls and notion of convergence reproduce the familiar ones on \(\mathbb{R}\). The word “space” refers to the underlying set together with its chosen metric; changing the metric can change what counts as close.

The first axiom says that distinct points have positive distance, and that a point has zero distance from itself. The second says that the distance does not depend on the order in which the two points are named. The third is the triangle inequality: going from \(x\) to \(z\) directly cannot be longer than going from \(x\) to \(y\) and then from \(y\) to \(z\).

An open ball collects all points whose distance from its center is less than the chosen radius. In the real line with its usual metric, this definition gives precisely the open intervals around the center.

Proof. First, \(|x-y|\geq0\), and \(|x-y|=0\) if and only if \(x-y=0\), which is equivalent to \(x=y\). Also,

For the triangle inequality, write \(x-z=(x-y)+(y-z)\). The triangle inequality for absolute value gives

Thus \(d\) satisfies all three metric axioms. Finally, for \(r>0\),

The last equivalence follows from the usual absolute-value inequality. Therefore \(B_d(x,r)=(x-r,x+r)\), as claimed. \(\square\)

Metric-Open Sets on the Real Line

A metric also specifies which subsets are open. A subset \(U\) of a metric space \((X,d)\) is called open if, for every \(x\in U\), there is some radius \(r>0\) such that \(B_d(x,r)\subseteq U\). The radius may depend on the point \(x\). This condition says that every point of \(U\) has some room around it that remains inside \(U\).

For the usual metric on \(\mathbb{R}\), the open-ball theorem makes this criterion familiar: a ball centered at \(x\) with radius \(r\) is the interval \((x-r,x+r)\). Consequently, a subset of \(\mathbb{R}\) is open in the metric sense exactly when every one of its points lies in an open interval contained in the subset. This is the usual definition of an open subset of the real line. Thus the metric does not introduce a different notion of openness here; it packages the familiar one in terms of distance.

A key detail in this argument is that the radius is allowed to depend on \(x\). Near an endpoint, the ball must be smaller than it would be near the middle of the interval. Requiring one fixed radius to work for every point would be a stronger condition and is not the definition of openness.

Convergence Expressed by the Metric

The same distance function expresses convergence. In a metric space, a sequence \((x_n)\) converges to \(x\) if, for every \(\varepsilon>0\), there is an \(N\in\mathbb{N}_0\) such that \(d(x_n,x)<\varepsilon\) whenever \(n\geq N\). In the real line with \(d(u,v)=|u-v|\), this is exactly the usual epsilon definition of convergence of a real sequence.

Proof. By metric convergence, for every \(\varepsilon>0\) there must be an \(N\) such that \(d(a_n,L)<\varepsilon\) for every \(n\geq N\). In the usual metric, \(d(a_n,L)=|a_n-L|\), so this condition is precisely that for every \(\varepsilon>0\), there is an \(N\) such that \(|a_n-L|<\varepsilon\) for every \(n\geq N\). That is the usual definition of \(a_n\to L\). The two definitions are therefore equivalent. \(\square\)

Why a Metric Gives at Most One Limit

The triangle inequality also guarantees that a sequence in any metric space cannot converge to two different points. This result does not depend on the real line or on the absolute-value formula; it follows from the metric axioms alone. It is a basic safeguard when identifying a limit from estimates.

Proof. Suppose \(x_n\to x\) and \(x_n\to y\). Let \(\varepsilon>0\). By convergence to \(x\), there is an \(N_1\) such that \(d(x_n,x)<\varepsilon/2\) whenever \(n\geq N_1\). By convergence to \(y\), there is an \(N_2\) such that \(d(x_n,y)<\varepsilon/2\) whenever \(n\geq N_2\). Set \(N=\max\{N_1,N_2\}\). For \(n\geq N\), the triangle inequality gives

This holds for every \(\varepsilon>0\). Since \(d(x,y)\geq0\), the only nonnegative number that is smaller than every positive \(\varepsilon\) is \(0\). Thus \(d(x,y)=0\), and the metric axiom that distance is zero only for identical points gives \(x=y\). \(\square\)

What the Metric Viewpoint Adds

On \(\mathbb{R}\), the usual metric records the same differences that already appear in sequence proofs. Its value is not that it changes those calculations, but that it gives a common language for several ideas: closeness is expressed by distance, neighborhoods by open balls, and convergence by eventual membership in every ball around the proposed limit. The open-ball theorem confirms that this language agrees with familiar intervals on the real line.

A common pitfall is to treat the word “distance” as if it always meant ordinary absolute difference. The metric axioms, not a particular formula, are the essential structure. In this tutorial the underlying set is \(\mathbb{R}\) and the chosen metric is \(d(x,y)=|x-y|\), so the usual notions are recovered. When another metric is chosen—even on the same underlying set—the sizes and shapes of balls, and potentially the resulting notions of closeness and convergence, must be checked from that metric rather than assumed.

For the real line with its usual metric, the translation is especially direct: replace \(|x-y|\) by \(d(x,y)\), and open intervals centered at \(x\) by balls \(B_d(x,r)\). The metric axioms then supply reusable tools, particularly the triangle inequality and the guarantee that limits are unique.