In the real numbers R prove that any Cauchy sequence has a limit point in the re
ID: 2941661 • Letter: I
Question
In the real numbers R prove that any Cauchy sequence has a limit point in the reals.
Explanation / Answer
If r1, r2, r3, ... is a Cauchy sequence of real numbers, then, if you pick any positive d, no matter how small, I will be able to find an index of that sequence, n, such that |rj - rk| = d 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 d. If you look at it another way, it means that for each rj in the sequence, there is a dj that is the minimum length into which all the terms in the sequence beyond rj will fit. As the index, j, gets very big, the corresponding dj's get closer and closer to zero. Now remember that every real number is a Cauchy sequence of rational numbers. So each of the rj's in the sequence of real numbers is itself a Cauchy sequence of rational numbers. r1 = q1,1, q1,2, q1,3, ... r2 = q2,1, q2,2, q2,3, ... r3 = q3,1, q3,2, q3,3, ... . . . . . . . . . . . . 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 rj in the sequence of real numbers I will pick a q from the ith row of the table above to be qi of a new sequence of rational numbers. Here's how to do it. Recall that for each ri there is a di into which all the r's beyond will fit. And because ri = qi,1, qi,2, qi,3, ... is a Cauchy sequence of rationals, there is an index, n, such that |qi,j - qi,k| = di whenever both j > n and k > n. In other words, however small di 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 di. And any one of those beyond that point (and in that same row) will do as my choice for qi. 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 |qj - qk| = dn whenever both j > n and k > n. Since the dn'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 qj must fall within dj of rj. Which means that |qj - rj| 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.