Schur's Theorem Proof Using Pigeonhole

We give a self-contained proof of Schur's theorem on monochromatic solutions of x + y = z, using nothing beyond the pigeonhole principle. The argument constructs a nested family of subsets of the positive integers. At each stage a base point (a "pin") is selected, and only those points lying at a distance of one fixed colour from that pin are retained; the pin then carries that colour as a label. After r+1 stages, two labels must coincide, and the corresponding pins together with any surviving point yield a monochromatic triple. Both the infinite form of the theorem and the finite form, with the explicit bound S(r) < (r^(r+2) − 1)/(r − 1), follow from the same construction — the only difference being whether the pigeonhole principle is applied in its infinite or its counting form. No appeal to Ramsey's theorem or to a compactness argument is required. The note is expository rather than original. The author arrived at the argument independently, and the final section records its relationship to Schur's own 1916 proof (Jahresbericht der Deutschen Mathematiker-Vereinigung 25, 114–117), which proceeds along substantially the same lines, as well as to the standard modern treatment via Ramsey's theorem applied to the colouring c({a,b}) = χ(|a−b|). A remark indicates how the bound can be sharpened to order r!·e within the same method.