12 Proof:For a positive e,define the e-neighborhood about 0o to be Θ(e)={9:Ie-0ol‖<} We want to show that Pa[0(Xn)∈Θ(e)】→1 asn→o.SinceΘ(e)is an open set,.we know that日nΘ(e)Cisa compact set (Assumption 2).Since Qo(0)is a continuous function (Assumption 3),then sup(o()}is a achieved for a 0 in the compact set.Denote this value by 0*.Since 0o is the unique max,let Qo(00)-Qo(0*)=6>0.12 Proof: For a positive , define the -neighborhood about θ0 to be Θ() = {θ : θ − θ0 < } We want to show that Pθ0 [ ˆ θ(Xn) ∈ Θ()] → 1 as n → ∞. Since Θ() is an open set, we know that Θ ∩ Θ()C is a compact set (Assumption 2). Since Q0(θ) is a continuous function (Assumption 3), then supθ∈Θ∩Θ()C {Q0(θ)} is a achieved for a θ in the compact set. Denote this value by θ∗. Since θ0 is the unique max, let Q0(θ0) − Q0(θ∗) = δ > 0.