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 its late terms bunch together: past some N, every pair sits within any given tolerance. Showing terms get pairwise close, without naming a limit, proves a sequence is Cauchy. That is the point of the criterion: it certifies convergence while the limit is still unknown. For 1 over n, tolerance 0.01 needs N equal to 100.
The approximations 1, 1.4, 1.41, 1.414, and so on bunch beautifully while chasing root 2 from inside the rationals. The sequence is Cauchy in the rationals yet has no rational limit. In the reals, completeness closes that gap, so Cauchy and convergent mean the same thing.
Bolzano-Weierstrass says every bounded real sequence has a convergent subsequence. The alternating (-1) to the n diverges, yet its even indexed terms sit constant at 1. Extracting such a subsequence is a skill: halve the trap, keep the infinite half, and repeat. Partial sums show the same idea when disturbances fade, as with 1 plus 1 over 2 plus 1 over 4 plus 1 over 8 reaching 1.875.
Cauchy means late terms bunch, completeness turns bunching into convergence, and boundedness always hides a convergent subsequence.
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8 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.