Q: Show ℓ¹ is Banach. Why isn't c₀₀ complete? ℓ¹: absolutely convergent series converge (coordinate-wise Cauchy → limit in ℓ¹!)! Take Cauchy (xⁿ): each coordinate Cauchy in ℝ: limit x. Show x ∈ ℓ¹, xⁿ → x: BANACH ✓! c₀₀ (finite sequences): xⁿ = (1,1/2,...,1/n,0,...): Cauchy, limit (1/k) ∉ c₀₀! Not complete! (Completion: ℓ¹!)