For the simple regression model y i = + i , i N[0, 2 ], prove that the sample mean

For the simple regression model y_i = μ + ε_i, ε_i ∼ N[0, σ²], prove that the sample mean is consistent and asymptotically normally distributed. Now consider the alternative estimator μ̂ = Σ_i w_i y_i ,w_i = i/(n(n+1)/2) = i/Σ_i i. Σ_i w_i = 1. Prove that this is a consistent estimator of μ and obtain its asymptotic variance.