I am currently studying goodness of fit tests and the $\chi^{2}$ distribution. To calculate the goodness of fit of a theoretical distribution, we compute the quantity $X^{2}=\sum_{i=1}^{n}\frac{(O_{i}-E_{i})^{2}}{E_{i}}$ Where $n$ is the number of outcomes, $O_{i}$ is the number of times the $i^{th}$ outcome is observed, and $E_{i}$ is the number of times we expect it to occur (given our distribution).
My book then states $X^{2} \sim \chi^{2}_{\nu}$ approximately, where $\nu$ is an appropriate number of degrees of freedom for the data.
It then defines the $\chi^{2}$ distribution as follows:
If $Z_{i} \sim N(0,1)$ and $X=\sum_{i=1}^{\nu}Z_{i}^2$, then $X\sim \chi^{2}_{\nu}$.
It also briefly proves that $X^{2}$ has an approximately $\chi^{2}$ distribution when $\nu=1$. My question is this:
How do we prove that, for any $\nu$, $X^{2}$ has an approximately $\chi^{2}$ distribution?
Intuitively I feel like this is the start of an induction proof, but I don't know how to show the inductive step. If this is the case, please show me how.
NOTE: I have just noticed that there is an added condition: Each of the $E_{i}$ are greater than or equal to 5.