An estimator $\hat\theta_n$ is said to be squared error consistent for $\theta$ if lim$_{n→ \infty} E[(\hat\theta_n - \theta)^2] = 0$
a) Show that any squared error consistent $\theta_n$ is asymptotically unbiased.
attempt: Let $E[(\hat\theta_n - \theta)^2] = E[\hat\theta^2_n - 2\theta\hat\theta_n + \theta^2] = E(\hat\theta^2_n) - 2\theta E(\hat\theta) + \theta^2 = E(\hat\theta^2_n) - [E(\theta_n)]^2 + [E(\theta_n)]^2 - 2\theta E(\hat\theta) + \theta^2 = Var(\hat\theta) + E[(\hat\theta) - \theta)]^2 $
Can someone please help me? I don't really know if this is a way to do it. Any help would be really appreciate it. Thank you in advance.