Proof of the Existence of a Real Limit to Every Cauchy Sequence of Real Numbers

Reminder that this page is optional material and you should spend only your spare time (if any) trying to understand it. It may be that this proof will be too difficult for you until you have gotten through the section on limits.

What We are Going to Prove

In the main text we discussed how a Cauchy sequence of real numbers is really a Cauchy sequence of Cauchy sequences. And each of those sequences is a sequence of rational numbers. Here we will give rigorous proof that a Cauchy sequence of real numbers does indeed converge to a real number. And we will do so by constructing a Cauchy sequence of rational numbers that converges to the same real number.

Reminder of What is Meant by a Cauchy Sequence

Remember the baseball pitching analogy. If

   r  =  q₁, q₂, q₃, ...

is a Cauchy sequence, then if you pick any positive value, δ, no matter how small, then I can show you how deep into the sequence you need to go so that every q beyond that point falls within a range that is no bigger than δ. For example, suppose the sequence is

  1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, ...

where each new entry the square root of 2 carried out to one more digit. If you tell me that δ is one millionth, I would tell you to go out six terms of the sequence. You will find that all of the terms beyond that fall into a space on the number line that is no greater than one millionth of a unit long. And even if you gave me a smaller δ, I would still be able to show you how deep into the sequence you had to go to have every term beyond that fit into a space no more than δ long on the number line.

In the main text we defined a real number as any Cauchy sequence of rational numbers. We also showed that you can take the difference of two real numbers by taking the difference of their Cauchy sequences term by term. And for the record, you can take the absolute value of any real number by taking the absolute value of its Cauchy sequence term by term. You can find the distance between two real numbers by taking the absolute value of their difference. We will use that distance formula several times in what follows.

Proof

   r₁, r₂, r₃, ...

is a Cauchy sequence of real numbers, then, if you pick any positive δ, no matter how small, I will be able to find an index of that sequence, n, such that

   |r_j - r_k|  ≤  δ

whenever both j > n and k > n. That is just another way of saying that I can go deep enough into the sequence to make all the terms beyond fit within a length of δ.

If you look at it another way, it means that for each r_j in the sequence, there is a δ_j that is the minimum length into which all the terms in the sequence beyond r_j will fit. As the index, j, gets very big, the corresponding δ_j's get closer and closer to zero.

Now remember that every real number is a Cauchy sequence of rational numbers. So each of the r_j's in the sequence of real numbers is itself a Cauchy sequence of rational numbers.

  r₁  ≡  q_1,1, q_1,2, q_1,3, ...

  r₂  ≡  q_2,1, q_2,2, q_2,3, ...

  r₃  ≡  q_3,1, q_3,2, q_3,3, ...
  .       .     .     .
  .       .     .     .
  .       .     .     .

where all the q's are rational numbers. We prove the theorem by constructing a sequence of rational numbers out of the ones in the table above that converges to the same real number as the r₁, r₂, r₃, ... sequence does.

For each r_j in the sequence of real numbers I will pick a q from the ith row of the table above to be q_i of a new sequence of rational numbers. Here's how to do it. Recall that for each r_i there is a δ_i into which all the r's beyond will fit. And because

  r_i  ≡  q_i,1, q_i,2, q_i,3, ...

is a Cauchy sequence of rationals, there is an index, n, such that

   |q_i,j - q_i,k|  ≤  δ_i

whenever both j > n and k > n. In other words, however small δ_i is, you can go deep enough into the q's in the ith row of the table so that all the q's to the right of that (and in that same row) will fall within a length of the number line no longer that δ_i. And any one of those beyond that point (and in that same row) will do as my choice for q_i.

Now think about the sequence of rational numbers, q₁, q₂, q₃, ... that we construct this way. Remember that q₁ comes from the first row of the table, q₂ from the second row of the table, and so on. And the way we have chosen them, it is true that

   |q_j - q_k|  ≤  δ_n

whenever both j > n and k > n. Since the δ_n's get closer and closer to zero as n gets very large, it must be that the q sequence we have constructed is indeed a Cauchy sequence. Not only that, but q_j must fall within δ_j of r_j. Which means that |q_j - r_j| gets closer and closer to zero as j gets very big. That is another way of saying that the q sequence we constructed converges to the same real number as the r sequence we started with. And that completes the proof.

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