Assume: $\eta, \epsilon_1{,}...,\epsilon_n$ are independent with uniformly bounded variance.
Define: $$q_j=\eta + \epsilon_j, \qquad j=1,\ldots,n$$
and $$\overline{q_{-i}}=\frac{1}{n-1}\sum_{j \not=i}^n q_j, \qquad i=1,\ldots,n$$
Now by the strong law of large numbers, $\overline{q_{-i}}$ goes almost surely to $\eta$. Why is this the case? Thanks!
Here is an image of the proof i try to understand:
