How did Georg Cantor’s work on trigonometric series lead him to study sets?

Georg Cantor’s transition into the study of sets was a direct result of his research into trigonometric series.1

In 1870, while building upon the work of Bernhard Riemann, Cantor demonstrated that a function could be represented by a trigonometric series in only one way. To further his investigation, he needed to consider the collection of points (numbers) that would not conflict with such a representation.1

This requirement led him to:
* Define irrational numbers in terms of convergent sequences of rational numbers in 1872.1
* Begin his foundational work on the theory of sets and the concept of transfinite numbers.1

As he analyzed these collections, Cantor became increasingly occupied with the properties of infinite sets of real numbers. By applying the device of one-to-one correspondence, he discovered that infinite sets could differ in size, ultimately leading to his revolutionary conclusion that some infinities are strictly larger than others.2341