Page "Proof that e is irrational" Paragraph 3

This is Joseph Fourier's proof by contradiction.

Initially e is assumed to be a rational number of the form a / b. We then analyze a blown-up difference x of the series representing e and its strictly smaller partial sum, which approximates the limiting value e. By choosing the magnifying factor to be the factorial of b, the fraction a / b and the partial sum are turned into integers, hence x must be a positive integer.

However, the fast convergence of the series representation implies that the magnified approximation error x is still strictly smaller than 1.

From this contradiction we deduce that e is irrational.