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Computable sequence of rationals with a noncomputable limit

Genady
in Analysis and Calculus

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This is a proof of a statement in Stillwell's "Reverse Mathematics", p.77:

Screenshot 2026-01-20 120557.png

My question is, how do we know that the sequence [math]r_1, r_2, ...[/math] converges?

  • Author

The sequence [math]r_n=\sum_{i=1}^n 2^{-f(i)}[/math] is increasing and bounded, thus it converges.

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