Let x1 xn be iid from a continuous cdf f on r and x0 be a

Question

Let X₁, ..., X_n be i.i.d. from a continuous c.d.f. F on R and X0 be a future observation that is independent of X_i's and has the c.d.f. F. Suppose that F is strictly increasing in a neighborhood of F^-1(α/2) and a neighborhood of F^-1(1 - α/2). Let F_n be the empirical c.d.f. defined. Show that the prediction interval C(X) = [F^-1_n (α/2), F^-1_n (1 - α/2)] for X₀ satisfies P(X₀∈ C(X)) → 1 - α, where P is the joint distribution of (X₀, X₁, ..., X_n).