Cauchy Sequences and the Meaning of Completeness

Closeness without a target

Every test so far needed the limit L up front. A Cauchy sequence removes that: (a_n) is Cauchy if its terms get close to each other. Precisely — for every epsilon > 0 there exists N such that for all m, n > N we have |a_m - a_n| < epsilon. No L appears anywhere. You are only asked whether the tail bunches up.

Two quick facts establish the easy direction. First, every convergent sequence is Cauchy: if a_n -> L, then by the triangle inequality |a_m - a_n| <= |a_m - L| + |L - a_n|, and both halves are small past some N. Second, every Cauchy sequence is bounded, by the same one-step argument that bounded convergent sequences. So Cauchy sits between “convergent” and “bounded.” The deep question is the reverse first arrow.

Why Cauchy sequences converge in R

Here is the theorem that crowns the track: in the real numbers, every Cauchy sequence converges. The proof is a beautiful collaboration of everything before it. A Cauchy sequence is bounded; by Bolzano–Weierstrass it has a subsequence converging to some L; and the Cauchy condition then forces the WHOLE sequence to that same L. Read it line by line.

Theorem: every Cauchy sequence (a_n) in R converges.

Step 1 (bounded). A Cauchy sequence is bounded.
Step 2 (a candidate limit). By Bolzano-Weierstrass, some subsequence
        a_{n_k} -> L for some real L.
Step 3 (the whole sequence catches up). Let e > 0.
   Cauchy: choose N so that |a_m - a_n| < e/2 for all m, n > N.
   Subsequence: choose k with n_k > N and |a_{n_k} - L| < e/2.
   Then for every n > N, using the triangle inequality:
       |a_n - L| <= |a_n - a_{n_k}| + |a_{n_k} - L|
                 <    e/2        +     e/2      = e.
   Hence a_n -> L.  QED

The Cauchy condition does the heavy lifting in step 3:
it lets ONE good subsequence term L drag the whole tail along.

Cauchy + bounded + Bolzano–Weierstrass = convergence. The subsequence finds the limit; the Cauchy condition spreads it to every term.

Completeness is exactly this

That the implication holds is the meaning of completeness. The completeness of the reals can be stated as: every Cauchy sequence of real numbers converges to a real number. There are no gaps for a bunched-up sequence to fall into. The rationals fail this — that is why we needed the reals at all.

The rationals are NOT complete -- a Cauchy sequence with no rational limit.

Define rationals by truncating the decimal expansion of sqrt(2):
  a_1 = 1.4,  a_2 = 1.41,  a_3 = 1.414,  a_4 = 1.4142, ...
Each a_n is rational. For m > n the terms agree to n decimals, so
  |a_m - a_n| <= 10^{-n}  ->  0,  so (a_n) is Cauchy.
In R it converges to sqrt(2). But sqrt(2) is irrational, so inside Q
the sequence bunches up around a HOLE -- it has no rational limit.

Moral: Cauchy means "trying to converge." Whether it succeeds
depends on the space. R fills every such hole; Q does not.

The same Cauchy sequence converges in R but has nowhere to land in Q. Completeness is the promise that R has no such holes.

This is the doorway out of the line. A complete metric space is any space in which every Cauchy sequence converges, and the Cauchy criterion becomes the standard tool for proving convergence when no candidate limit is in sight — in function spaces, in real analysis proper, and far beyond. You have now built, from epsilon and N, the single idea on which most of analysis rests.