How about Bolzano's 1817 proof of the intermediate value theorem?
In English here: Russ, S. B. "A Translation of Bolzano's Paper on the Intermediate Value Theorem." Hist. Math. 7, 156-185, 1980.
Or in the original here: Bernard Bolzano (1817). Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation. In Abhandlungen der koniglichen bohmischen GeseUschaft der Wissenschaften Vol. V, pp.225-48.
Not fully rigorous, according to today's standards, but perhaps his method of proof could be considered a breakthrough nonetheless.
