Slow Cauchy sequences allow most one-dimensional real analysis to be carried out in weak systems conservative over RCA0, equating results like Bolzano-Weierstrass to the infinite pigeonhole principle instead of arithmetical comprehension.
The cohesive principle and the Bolzano-Weierstraß principle
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Avoiding logical strength in real analysis
Slow Cauchy sequences allow most one-dimensional real analysis to be carried out in weak systems conservative over RCA0, equating results like Bolzano-Weierstrass to the infinite pigeonhole principle instead of arithmetical comprehension.