I have the following equation from some linear regression model where $x_i$ and $u_i$ are random variables: $$ b=β+(\sum^N_{i=1}x_i x_i')^{-1} \sum^N_{i=1}x_iu_iγ $$ after applying $plim$ to the equation, my textbook tells me that we get:
$$ plim\ b=β+ (\sum^N_{i=1}x_i x_i')^{-1}E\{x_iu_i\}γ $$
while the first part of the equation is clear to me I don't understand how the $plim$ of $\sum^N_{i=1}x_iu_iγ $ converges to $E\{x_iu_i\}γ $.
My intuition tells for convergence to the expected value we (actually) need an extra factor of $1/N$ as in: $$ plim \frac{1}{N}\sum^N_{i=1}x_iu_iγ = E\{x_iu_i\}γ $$
What am I getting wrong about the plim here?
It should be \begin{align} (\sum_{i=1}^N x_i x_i^T)^{-1} \sum_{i=1}^n x_i u_i \gamma &= \frac{N}{N}(\sum_{i=1}^N x_i x_i^T)^{-1} \sum_{i=1}^n x_i u_i \gamma \\ &= \left(\frac{1}{N}\sum_{i=1}^N x_i x_i^T\right)^{-1} \frac{1}{N}\sum_{i=1}^n x_i u_i \gamma\\ &\to \left(E\left[x_i x_i^T\right]\right)^{-1} E\left[ x_i u_i\right]\gamma \end{align} where $\to$ stands for convergence in probability.