Note that any point in an n-dimensional Euclidean space can be expressed as the linear combination of a set of orthoganal unit vectors. sin(wt),cos(wt),sin(2wt),cos(2wt),.... form a set of orthogonal "unit vectors" in a function space (where each point is a time function) and thus can be used to represent a periodic function. To represent a non-periodic function one then needs to use uncountably many basis functions, roughly speaking. This leads to the idea of FT. It is a natural thinking line. Although historically some good minds could not believe that almost any periodic function can be represented by sin's and cos's. The lesson is keep generalizing a cool idea until you hit a wall.