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Two theorems of elmentary arithmetic, one stating that the sum of thereciprocals of any number of consecutive positive integers is never an integer,and a generalization thereof by Trygve Nagell, are shown to be provable insidea very weak arithmetic, Richard Kaye's $PA^-$, in which there is no inductionaxiom whatsoever.

The sum of irreducible fractions with consecutive denominators is never an integer in a very weak arithmetic