We know that if we have $N-n$ gaussian iid RVs $\{e_i\}$ with mean $0$ and variance $1$, the RV $x = \sum e_i^2$ is $\chi^2$ distributed with $N-n$ degrees of freedom. We have $N$ larger than $n$.
I do not see why the Law of Large Numbers implies that
$\frac{x}{N} \to 1$ in probability as $N \to \infty$
I know that $\frac{\sum{e_i}}{N-n} \to 0$ in as $N \to \infty$
If $e_1,\ldots,e_N$ are i.i.d. with mean $0$ and variance $1$, then $e_1^2,\ldots,e_N^2$ are i.i.d. with mean $1$, so, by the law of large numbers, $\frac1N\sum\limits_{i=1}^Ne_i^2\to1$ as $N\to\infty$.