Cauchy sequence
A Cauchy sequence is one whose terms eventually huddle together, regardless of any external target. The test is purely internal: far enough out, any two terms are arbitrarily close to each other — not to some named limit, but simply to one another. It captures the idea of a sequence 'tightening up' without first having to know where it is headed.
Precisely: (a_n) is Cauchy if for every epsilon > 0 there exists N such that for all m, n > N we have |a_m - a_n| < epsilon. Contrast this with the epsilon-N definition of a limit, which compares a_n to a fixed L; here the comparison is between pairs of terms. This is its great virtue: you can certify Cauchyness without naming or even knowing the limit.
In the real numbers, Cauchy and convergent are equivalent — a sequence converges if and only if it is Cauchy. This equivalence is precisely the completeness of R, and it is what makes R a complete space. The subtlety is that the equivalence can fail elsewhere: in the rationals, a sequence of decimal truncations of sqrt(2) is Cauchy yet has no rational limit. The rationals are not complete, and that gap is exactly what the reals fill in.
The truncations 1, 1.4, 1.41, 1.414, 1.4142, ... of sqrt(2) form a Cauchy sequence: terms past index N agree to N decimal places. In R it converges (to sqrt(2)); viewed inside the rationals alone, it is Cauchy with no limit.
Cauchy without a limit is the signature of an incomplete space like the rationals.
It is not enough that consecutive terms get close — |a_{n+1} - a_n| -> 0 does not imply Cauchy. The partial sums of the harmonic series are a famous counterexample: gaps between neighbors shrink, yet the sequence diverges.