I have come across sequence of functions and numbers in the context of approximation theory and understand that a lot of theory of functional analysis came out with the idea to approximate solutions to partial differential equations.
In what contexts have people been curious to study/develop the theory of sequence of events - tossing of coin an infinite number of times or in some limit are not somethings I believe would motivate these kind of pursuits.
Do some mathematicians and others who use probability have a practical context to share in this kind of system modelling with properties of increasing, decreasing, countability, uncountability of sequences of events and their probabilities come into picture?
There are very many applications of stochastic processes. To name a few, gambling, statistical sampling, insurance, communication networks, stock option pricing.