Since S is the topological product of circles, it is compact.
A function of S is periodic along any dimension, with period 2π. Go all the way around the circle, and f must repeat. Thus f might be a wave form in time, or a pattern in 2 dimensional space.
Let W be the ring of functions spanned by 1, sin(nx) for each integer n and each coordinate x, and cos(nx) for each integer n and each coordinate x. The ring includes functions like sin(3x)cos(2y), and cos(y)sin(17z), but it need not contain sin(5x)sin(2x), or any multiple thereof, because sin(5x)sin(2x) is a linear combination of cos(7x) and cos(3x). Nor do we need cos(9x)2, as this comes from 1 and cos(18x). (These are standard trig identities.) So, if there are 3 coordinates x y and z, the functions of W are products of 0 1 2 or 3 trig functions, sines or cosines, acting on multiples of x y or z.
Let two points of S differ in their x coordinate. Suppose sin(x) and cos(x) fail to separate them. In other words, cos(x1) = cos(x2), and sin(x1) = sin(x2). The former assertion says x1 and x2 are symmetric about π. This is incompatible with the latter assertion. Therefore W separates points.
Apply the previous theorem, and any continuous function on the generalized torus, i.e. any continuous periodic function in Rn, can be approximated uniformly by sines and cosines at the fundamental frequency and its higher harmonics.
If you didn't understand this, that's ok. It's not as helpful as you might think. We really want a fourier series, and an approaching sequence of functions does not always imply a series. After all, if an approaching sequence of polynomials implied a power series, then every continuous function would be analytic, and differentiable, and that is not the case. The Holy Graille is a convergent fourier series. You can read more about that here.