"Bolzano points out that Gauss’s first proof is lacking in rigor; he then gives in 1817 a “purely analytic proof of the theorem, that between two values which produce opposite signs, there exists at least one root of the equation” (Theorem III.3.5 below). In 1821, Cauchy establishes new requirements of rigor in his fa- mous “Cours d’Analyse”. The questions are the following:

– What is a derivative really? Answer: a limit.

– What is an integral really? Answer: a limit.

– What is an infinite series a1 + a2 + a3 + . . . really? Answer: a limit.

This leads to

– What is a limit? Answer: a number.

And, finally, the last question: – What is a number?

Weierstrass and his collaborators (Heine, Cantor), as well as Me ́ray, answer that question around 1870–1872. They also fill many gaps in Cauchy’s proofs by clarifying the notions of uniform convergence (see picture below), uniform continuity, the term by term integration of infinite series, and the term by term differentiation of infinite series." (Analysis by it's History, Hairer).

And if you like pictures, here is an indirectly related one (http://en.wikipedia.org/wiki/Algebraic_number#The_field_of_algebraic_numbers).

– What is a derivative really? Answer: a limit.

– What is an integral really? Answer: a limit.

– What is an infinite series a1 + a2 + a3 + . . . really? Answer: a limit.

This leads to

– What is a limit? Answer: a number.

And, finally, the last question: – What is a number?

Weierstrass and his collaborators (Heine, Cantor), as well as Me ́ray, answer that question around 1870–1872. They also fill many gaps in Cauchy’s proofs by clarifying the notions of uniform convergence (see picture below), uniform continuity, the term by term integration of infinite series, and the term by term differentiation of infinite series." (Analysis by it's History, Hairer).

And if you like pictures, here is an indirectly related one (http://en.wikipedia.org/wiki/Algebraic_number#The_field_of_algebraic_numbers).

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