Tilings of convex sets by mutually incongruent equilateral triangles contain arbitrarily small tiles

We show that every tilingof a convex setin the Euclidean plane $\mathbb{R}^2$ by equilateral trianglesof mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of tilingsof the full plane $\mathbb{R}^2$, which is based on a surprising connection to a random walkon a directed graph.