Let a₁ ≠ a₂ be positive numbers, and b₁ = 1 / a₁. Define the sequence xₙ = (aₙ, bₙ) in ℝ² by aₙ₊₁ = 1/3 aₙ + 2/3 aₙ₋₁; bₙ = 1/5 (b₂ₙ₋₁ + 2). Show that {xₙ} is convergent.