I want to write "proof" that a ${\chi}^2$ statistic becomes larger and larger as the sample size increases. I have come up with the following:
For ${\chi}^2=\sum_{j=1}^{n} \frac{(O_j - E_j)^2}{E_j}\quad$ if $\quad \Pr(\vert O_{j}-E_{j}\vert>0)=\epsilon\quad$ for $\quad\epsilon>0$ then
$\lim_{n\to\infty}\sum_{j=1}^{n} \frac{(O_j - E_j)^2}{E_j}\rightarrow \infty$ almost surely.
Is this enough and/or correct or am I missing something?