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.
Errett Bishop. Foundations of constructive analysis. McGraw-Hill Book Company, New York, San Francisco, St. Louis, Toronto, London, and Sydney, 1967, xiii + 370 pp. - Errett Bishop. Mathematics as a numerical language. Intuitionism and proof theory, Proceedings of the summer conference at Buffalo N.Y. 1968, edited by A. Kino, J. Myhill, and R. E. Vesley, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam and London1970, pp. 53–71.
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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.