Bunching up: Cauchy sequences
Convergence asks the terms to approach a specific point L. But often we want to detect that the terms are settling without naming the destination. A sequence (x_n) is a Cauchy sequence if its terms eventually get and stay close to one another: for every epsilon > 0 there is N such that d(x_m, x_n) < epsilon for all m, n ≥ N. No limit appears in this definition — only the mutual distances among the terms.
Every convergent sequence is Cauchy — once all terms are within epsilon/2 of the limit, they are within epsilon of each other. The converse is the delicate part, and where spaces differ.
Claim: every convergent sequence is Cauchy.
Suppose x_n -> L. Fix epsilon > 0.
There is N with d(x_n, L) < epsilon/2 for all n >= N.
Now take any m, n >= N. By the triangle inequality,
d(x_m, x_n) <= d(x_m, L) + d(L, x_n)
< epsilon/2 + epsilon/2
= epsilon.
So the terms are mutually within epsilon for m, n >= N,
which is exactly the Cauchy condition. QED
The CONVERSE (Cauchy => convergent) can FAIL:
In X = the rationals Q with d(x, y) = |x - y|,
the decimal truncations 1, 1.4, 1.41, 1.414, ... are Cauchy
(consecutive terms differ by at most 10^(-k)),
but their would-be limit sqrt(2) is NOT in Q.
The sequence bunches toward a hole.Completeness
A metric space is a complete metric space when the converse holds everywhere: every Cauchy sequence actually converges to a point of the space. Completeness says the space has no holes — wherever the terms bunch, a genuine limit is waiting. The real numbers R are the model example; their completeness is essentially the content of the completeness axiom that distinguishes R from Q. The rationals Q are the model counterexample, as the √2 sequence above shows.
Filling the holes: completion
If a space has holes, can we plug them? Yes, and canonically. The completion of a metric space X is a complete space X-hat that contains a faithful copy of X as a dense subset — every point of X-hat is a limit of points of X. “Faithful copy” is made precise by an isometry: a map that preserves all distances exactly, d(f(x), f(y)) = d(x, y). The completion of Q under |x − y| is precisely R; every irrational is the limit of a Cauchy sequence of rationals.
The construction is elegant: take the set of all Cauchy sequences in X, declare two of them equivalent when the distance between their corresponding terms tends to 0, and let the new points be these equivalence classes. The √2 hole in Q literally becomes the class of all rational sequences that bunch toward it. This is the same idea that builds R from Q in the first place.