[1] The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.
[2] By contrast, the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point,[3] but it doesn't describe how to find the fixed point (see also Sperner's lemma).
This is clear when examining a sketched graph of the cosine function; the fixed point occurs where the cosine curve y = cos(x) intersects the line y = x. Numerically, the fixed point (known as the Dottie number) is approximately x = 0.73908513321516 (thus x = cos(x) for this value of x).
Don Zagier used these observations to give a one-sentence proof of Fermat's theorem on sums of two squares, by describing two involutions on the same set of triples of integers, one of which can easily be shown to have only one fixed point and the other of which has a fixed point for each representation of a given prime (congruent to 1 mod 4) as a sum of two squares.
Since the first involution has an odd number of fixed points, so does the second, and therefore there always exists a representation of the desired form.