Short answers, pulled from the story.
Euclid wrote the Elements around 300 BCE which contains Proposition X.1 describing the Method of exhaustion. This technique allowed mathematicians to approximate areas and volumes by getting closer to a true value without reaching it in finite steps.
Grégoire de Saint-Vincent provided the first formal definition of a limit for geometric series in his 1647 work Opus Geometricum. He described the terminus as an end point that no progression can reach even if continued into infinity yet one that can be approached nearer than any specific segment.
Bernard Bolzano developed the basics of the epsilon-delta technique in 1817 to define continuous functions while Augustin-Louis Cauchy followed up on this foundation in 1821. Karl Weierstrass later formalized the definition of the limit of a function known as the epsilon-delta definition of limit.
In a metric space with distance function d, the limit when it exists is an element l such that given epsilon there exists an N such that for each n greater than N we have d(sn, l) less than epsilon. In a topological space with topology T, the limit when it exists is a point x such that given an open neighborhood U of x there exists an N such that for every n greater than N, sn is in U.
A series which converges for all orderings is called unconditionally convergent and proven equivalent to absolute convergence. The Riemann series theorem states that depending on the ordering, partial sums of conditionally convergent series can be made to converge to any real number as well as positive or negative infinity.