Intro to Real Analysis, Cauchy sequences, and the definition of the Real Numbers

In order to any kind of analysis we create a metric space. A metric space is a set equipped with a distance function \(d(a,b)\to [0,\infty)\). We read this as “the distance between \(a\) and \(b\)”. This must satisfy \(d(a,b)=0 \Longleftrightarrow a=b\) and \(d(a,b)=d(b,a)\) as well as \(d(a,b)\leq d(a,c)+d(c,b)\). The first two conditions must clearly be true if we are to call this a distance function, but the last is less obvious. This is called the triangle inequality and it is used a lot. It basically means that our distance function finds the shortest path, and it must be the same length or more to go through a third point. This is obviously true if you think about it in 2d space:

We will now define what makes a sequence converge. Consider the sequence \(\{p_n\}\) which is the set \(\{p_1,p_2,p_3,\cdots\}\). We claim that this sequence converges to \(L\) (limit). Then for any \(\epsilon >0\), there exists an \(N\) such that for any \(n\geq N\), \(d(p_n,L)<\epsilon\).

This formalizes the idea of getting closer and closer to a value \(L\). Any distance \(\epsilon\) I give you, you can give me an index \(N\) after which everything is closer than \(\epsilon\). If I tell you \(\epsilon = 0.001\), you have to be able to give me a number \(N\), perhaps \(N=1000\), so that everything after that is closer than \(0.001\), meaning \(d(p_{1000},L)<.001\) and \(d(p_{45001}, L)0\), there exists \(N\) such that for any \(n,m\geq N\), \(d(p_n,p_m)<\epsilon\). It is important to distinguish this from the definition of convergence. Cauchy sequences have a special relationship with convergent sequences.

It can be proved that convergent sequences are Cauchy. This makes sense, since if elements of the sequence are getting closer to some limit, then they must get closer to each other. However, it is not true that Cauchy sequences all converge. How can this be? Well, I showed an example before, where the “limit” was not in the metric space. In a sense, the metric space is “missing” the value where the limit should be.

In some metric spaces, every Cauchy sequence converges. This is called “the Cauchy Criterion”. For example, in the metric space \([0,1]\), every Cauchy sequence converges. You’ve patched the holes where the limits should be by putting a limit there. In a sense, you’ve completed it. If a metric space passes the Cauchy Criterion, then it is complete. There are no fake limits which are missing from the space.

This concept is very powerful. Consider \(\mathbb{Q}\), the rational numbers. Are they complete? Consider the sequence \(\{3,3.1,3.14,3.141,3.1415,\cdots\}\) which only has rational elements. We know that this will eventually “converge” to \(\pi\), but \(\pi\) is not in \(\mathbb{Q}\) so the sequence does not converge at all.

In a sense we can define \(\mathbb{R}\), the real numbers, as the completion of the rationals. We can define it as the smallest metric space containing \(\mathbb{Q}\) for which the Cauchy Criterion is satisfied. We add in all the points which are the limit of some rational sequence. There are different ways to define the real numbers which I may discuss in the future, but this is a particularly neat one, I think.

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