Cauchy Sequences and the Bolzano-Weierstrass Theorem
Terms that eventually crowd together converge, in the reals but not in the rationals. Every bounded sequence has a convergent subsequence, which is the workhorse behind existence proofs.
What a learner can do afterwards
- Show a sequence is Cauchy without naming its limit
- Give a Cauchy sequence of rationals with no rational limit
- Extract a convergent subsequence from a bounded sequence
1 · Read
A sequence is Cauchy when late terms bunch together. Formally, for every positive epsilon there is an integer N such that for all m and n greater than or equal to N, the distance between a_m and a_n is less than epsilon. No limit appears in this definition. For 1 over n and tolerance 0.01, choose N equal to 101: two terms past that cutoff differ by at most 1 over 101, which is below 0.01. Cauchy sequences are bounded too. Use epsilon 1: every term after one cutoff lies within 1 of a fixed tail term, and the finitely many earlier terms can be covered by one larger bound.
The approximations 1, 1.4, 1.41, 1.414, and so on are rational decimal truncations that chase root 2. Once k decimal places are fixed, the truncation is within 10 to the negative k of root 2, so two sufficiently late truncations differ by less than any chosen tolerance. They form a Cauchy sequence in the rationals, but their real limit is irrational, so no rational limit exists. In the reals, completeness closes that gap: Cauchy and convergent mean the same thing.
Bolzano-Weierstrass says every bounded real sequence has a convergent subsequence. To extract one, trap the sequence in a closed interval, bisect the interval, keep a half containing infinitely many terms, and repeat. The nested intervals point to a limit for a chosen subsequence. The alternating (-1) to the n diverges, yet its even indexed terms stay constant at 1. Geometric partial sums show the same bunching idea: 1 plus 1 over 2 plus 1 over 4 plus 1 over 8 equals 1.875, and later additions get smaller.
Cauchy means late terms bunch, completeness turns bunching into convergence, and boundedness always hides a convergent subsequence.
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2 · Watch
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Where it sits
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.