Cauchy Sequences As Real Numbers!?

Introduce Cantor’s Construction of $\mathbb{R}$

Our understanding of the set of real numbers may derive from the durations of time and lengths in space. We think of the real line, or continuum, as being composed of an (uncountably) infinite number of points, each of which corresponds to a real number. I introduce one way to construct the real numbers, developed by a German mathematician Georg Cantor, by first talking about a famous version of Zeno’s paradox in the context of a race between Achiles (a legendary Greek warrior) and the Tortoise:

“Achilles gives the Tortoise a head start of, say 10 m, since he runs at 10 $ms^{-1}$ and the tortoise moves at only 1 $ms^{-1}$ . Then by the time Achilles has reached the point where the Tortoise started ( $T_0 = 10$ m), the slow but steady individual will have moved on 1 m to $T_1 = 11$ m. When Achilles reaches $T_1$ , the laboring Tortoise will have moved on 0.1 m (to $T_2 = 11.1$ m). When Achilles reaches $T_2$ , the Tortoise will still be ahead by 0.01 m, and so on. Each time Achilles reaches the point where the Tortoise was, the cunning reptile will always have moved a little way ahead.”

If we think of the distances Achilles has to travel, first 10 m to $T_0$ , from 1 m to $T_1$ , and from 0.1 m to $T_2$ …, we can write it as a sum of a Geometric Series:

$10 + 1 + 0.1 + .... + 10^{2-n} +$ …

Since the distance that Achilles travels to catch the tortoise is the sum of a geometric series where $|r| < 1$ , we know that the distance is finite as the series eventually converges. So Zeno’s argument is based on the assumption that you can infinitely divide space (the race track) and time (how long it takes to run).

Lots of mathematicians have disputed whether the continuum can be represented as a set of points, until Cantor’s work on the construction of $\mathbb{R}$ was introduced and recognized.

Before we go into Cantor’s work, I want to quickly review what Abbott discusses in Chapter 2 of our textbook. We define the convergence of a sequence $(a_n)$ to a real number a if (by definition) for every $\epsilon > 0$ , there exists an $N \in \mathbb{N}$ so that whenever $n \ge N$ , it follows that $|a_n-a| < \epsilon$ . We also define a Cauchy Sequence $(a_n)$ if (by definition) for every $\epsilon > 0$ , there exists an $N \in \mathbb{N}$ so that whenever $m,n \ge N$ , it follows that $|a_n-a_m| < \epsilon$ . The Cauchy Criterion then guarantees that a sequence converges if and only if it’s a Cauchy sequence. Note that a real number may have more than one rational Cauchy sequence converging to it, so here’s when equivalence relation comes in.

Cantor gave his construction of the real numbers as equivalence classes of Cauchy sequences. We let $\mathcal{C}$ be the set of all Cauchy sequences in $\mathbb{Q}$ . Let $(a_n)$ and $(b_n)$ be elements in $\mathcal{C}$ , then we can define an equivalence relation such that $(a_n) \sim (b_n)$ if and only if $|a_n-b_n| \rightarrow 0$ . However, we need to prove rigorously that such an equivalence relation exists.

In order to show a relation is an equivalence relation, we need to show reflexivity, symmetry, and transitivity.

Reflexivity: $a_n-a_n=0$ , we then have the sequence with constant terms of 0 converges to 0, so $(a_n) \sim (a_n)$ and the relation is reflexive.
Symmetry: Assume $(a_n) \sim (b_n)$ , then $|a_n-b_n| \to 0$ . We then have $|b_n-a_n| = |-(a_n-b_n)| = |a_n-b_n| \to 0$ , so $(b_n) \sim (a_n)$ and thus symmetry holds.
Transitivity: Assume $(a_n) \sim (b_n)$ and $(b_n) \sim (c_n)$ . So $|a_n-b_n| \to 0$ and $|b_n-c_n| \to 0$ . Then according to the definition of convergence, for $\epsilon > 0$ , there exists an N in natural numbers such that for all n > N, $|a_n-b_n| < \dfrac{\epsilon}{2}$ , and $|b_n-c_n| < \dfrac{\epsilon}{2}$ . Then by Triangle Inequality,

$|a_n-c_n| = |(a_n-b_n) + (b_n-c_n)| \le |a_n-b_n|+|b_n-c_n| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$ .

We then see that for n > N, we have $|a_n-c_n| < \epsilon$ , which indicates that $|a_n-c_n| \to 0$ and that $(a_n) \sim (c_n)$ .

Now we’ve shown that we actually have an equivalence relation on $\mathcal{C}$ , and we now define the set of real numbers $\mathbb{R}$ as the equivalence classes $(a_n)$ of Cauchy sequences of rational numbers. That is, each such equivalence class is a real number!

Fill in the Gaps

Throwing back to ancient times, it was known that there exist numbers representing a length in space which cannot be represented as a rational number. These numbers can be seen as “gaps” on the continuous number line which are not filled by rationals. This leads us to an intuitive understanding of completeness – “The real number line is complete means that there are no such gaps which are not covered by a real number”.

Now we’ve constructed a set of equivalence classes which may be the set of real numbers, but we’d better be careful here – the set of real numbers we know has well-defined binary operations such as addition and multiplication, and the real numbers should also follow properties such as the Archimedean Property, the Density of $\mathbb{Q}$ in $\mathbb{R}$ , and boundedness. So there are some fun works to do in order to show that the one we just constructed would actually satisfy, which we won’t cover in the current blog.

Moreover, we’ve learned from class that every Cauchy sequence of real numbers converges. Now, one needs to be very careful to unpack exactly what this means in the terms of the construction of Cantor. Each real number is itself an equivalence class of Cauchy sequences of rational numbers, so, if you take a sequence of real numbers $(x_n)$ , you have to realize that $x_1 = (a_n)$ , $x_2 = (b_n)$ , $x_3 = (c_n)$ , etcetera. Then, we would use the binary subtraction of the equivalence classes of Cauchy sequences of rational numbers in order to define what it means to be Cauchy in this case, and finally we would have to show that this sequence of equivalence classes must converge to another equivalence class! It may sound a bit crazy, because we’ve got sequences of sequences and convergence to another sequence, but it turns out to work well and fill all the gaps that existed in the field $\mathbb{Q}$ .

Some Historical Context

Cantor initially put forward his construction of the real numbers in 1872, which was absolutely a strike. However, before that, Dedekind, another German mathematician that Cantor met on his honeymoon, provided the first rigorous construction of $\mathbb{R}$ through cuts, by separating all the real numbers in a series of two parts so that each real number in one part is less than every real number in the other.

Approaching from a different means, Cantor’s innovative ideas were not received well in the first place. He encountered resistance, sometimes fierce resistance, from people such as his old professor Kronecker and Poincaré, as well as from philosophers like Wittgenstein and even from some Christian theologians, who saw Cantor’s work as a challenge to their view of the nature of God. But Cantor’s ideas eventually went through all of these obstacles and became part of the mainstream in mathematics, which later contributed tremendously in the field of physics and metaphysics.

Posted on October 5, 2016 by Han Gao
Categories: Fall 2016

Math 320: Real Analysis

Cauchy Sequences As Real Numbers!?

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