Calculating probability limit

Question

I tried taking conditional probability on $\epsilon,$ to change the question in a form where we are taking plim of $\mu^2$ plus some noise. However, I'm having difficulties showing the noise part formally. Thank you in advance![enter image description here][1]

Question: Suppose $(x_i , \epsilon_i)$ ~ iid with $E(x_i) = \mu, var(x_i)= \sigma^2$, and $E[\epsilon_i|x_i]=x_i,$ Find the $lim_{n\rightarrow \infty}$ $n^{-1} \sum\limits_{i=1}^n x_ie_i$

Answer 1

Hint

If $\ (x_i,\epsilon_i)\ $ are independent and identically distributed, then so are $\ x_i\epsilon_i\ $, and \begin{align} E\big[x_i\epsilon_i\big]&=E\big[E[x_i\epsilon_i|x_i]\big]\\ &=E\big[x_iE[\epsilon_i|x_i]\big]\\ &=E\big[x_i^2\big]\\ &=\sigma^2+\mu^2\ . \end{align} The strong law of large numbers will therefore give you your limit.