Real Analysis 7 | Cauchy Sequences and Completeness [dark version]

The core takeaway is that in the real numbers, “Cauchy” behavior and convergence are the same thing—and that equivalence unlocks practical convergence tests built on supremum/infimum. Instead of tracking a sequence’s distance to a specific limit value (which presupposes knowing the limit), a Cauchy sequence is defined by internal consistency: its terms eventually get arbitrarily close to each other. For real-number sequences, this internal closeness is enough to guarantee the sequence actually converges, because the real line has no “holes” (the completeness axiom).

The transcript starts by contrasting the usual definition of convergence with the Cauchy idea. Convergence requires a number a such that |a − a_n| becomes smaller than any ε after some index N. Cauchy sequences avoid introducing that unknown a. They require that for every ε > 0, there exists N such that for all indices n, m ≥ N, the distance between terms satisfies |a_n − a_m| < ε. Such sequences are called Cauchy sequences (named after the property, not after a limit). The key fact emphasized is that for sequences of real numbers, being Cauchy is equivalent to being convergent—so one can use whichever definition is more convenient.

From there, the discussion turns to completeness in a set-theoretic form: Dedekind completeness. If a subset M of the real numbers is bounded above, then its supremum exists as a real number; similarly, if M is bounded below, its infimum exists. The proof sketch for the supremum uses a bisection-style construction. Starting with an upper bound B1 and an element A1 in M, the midpoint C1 = (A1 + B1)/2 is examined. If C1 remains an upper bound, it becomes the new upper bound; if not, a larger element from M replaces A1 while B1 stays. Repeating this recursively generates sequences (A_n) and (B_n) whose B_n values remain upper bounds and tighten toward the supremum.

To show this construction works, the transcript argues that the “gap” between successive bounds shrinks by halving at each step. Quantitatively, the distance between B_n and B_m (for m > n) can be bounded by a term like (1/2)^(n−1) times the initial gap (B1 − A1). That shrinking gap implies (B_n) is a Cauchy sequence. Completeness then forces (B_n) to converge, and the only plausible limit is the supremum of M. The same logic applies to infimum by symmetry.

Finally, the transcript uses these supremum/infimum results to produce convergence criteria. A monotonically decreasing sequence that is bounded from below must converge. Dually, a monotonically increasing sequence that is bounded from above must converge. These tests are presented as easier to check than the direct ε-definition of convergence, and they rely on the existence of supremum or infimum to supply the needed limit behavior.