Spurious regression help

Hi guys,
I am struggling with a spurious regression problem. I have attached the question and my working so far (the rest of my working is in the top comment):
I have managed to do most of it but cannot work out how to calculate what the variance of the estimator u_t converges/diverges to. I'm also not sure on why my assumptions are the way they are (I always assumed assumptions were to make the maths easier but I wouldn't know how to explain them) and I am not sure if I have done the correct thing to find the limit of a(hat).
Thanks in advance for any help


No cake for spunky
I have what may be a totally off topic question. You are predict Y with a lagged value of Y. Yt =Yt-1 plus an error term. And using OLS to do that. I thought you could not do that, it resulted in biased indicators. Perhaps this is only true with time series approaches.