Learn Mathematics With The Right Guidance

Step 1 (Total Boundedness): Let ε > 0. The family of open balls {B(x, ε) : x ∈ X} forms an open cover of X. By compactness of X, there exists a finite subcover {B(x₁, ε), B(x₂, ε), ..., B(xₖ, ε)}. Hence, X is totally bounded.

Step 2 (Cauchy Sequences): Let (xₙ) be any Cauchy sequence in X. Since X is compact, (xₙ) possesses a subsequence (xₙ_k) converging to some limit point x* ∈ X.

Step 3 (Convergence of Full Sequence): Given ε > 0, choose N such that d(xₙ, x_m) < ε/2 for all n, m ≥ N, and d(xₙ_k, x*) < ε/2. Then d(xₙ, x*) ≤ d(xₙ, xₙ_k) + d(xₙ_k, x*) < ε. Thus xₙ → x*.