Cauchy Implies Convergent in ℝ

Real Analysis · Axiom Academy

LESSON Cauchy Implies Convergent in Why a sequence whose terms bunch together must land somewhere — completeness fills in the limit that leaves out. A sequence (a_n) is Cauchy if, no matter how small a tolerance you name, there is a cutoff N beyond which every pair of terms is within of one another. There is no limit in the definition — only the terms talking to each other. An internal condition — it never names a destination Picture it on the number line. Shrink and the matching tail fits inside an ever-smaller window. Those windows nest and close down on a single point — and on a complete line , that point exists and is the limit. 2. Completeness Supplies the Limit Here is the whole argument, the one place completeness does the work. Watch the three links: a Cauchy sequence is bounded , so Bolzano–Weierstrass pulls out a subsequence that converges to some , and then being Cauchy forces the entire sequence onto that same L . 3. The Same Sequence, Stranded in Completeness is not free — it is exactly what lacks. Run the Babylonian iteration . Every term is rational, the terms bunch together (it is Cauchy), yet the point they close in on is — and . The shrinking tails close on , which contains. The sequence converges. The completeness axiom delivered the limit. Same Cauchy sequence, but . The tails close on a gap, so there is no rational limit. Cauchy is not enough here.

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