Prove the Johnson–Lindenstrauss lemma

Prove the Johnson–Lindenstrauss lemma from first principles: for any $x_1,\dots,x_N\in\R^D$ and $\eps\in(0,1)$, if $A\in\R^{k\times D}$ has i.i.d. $\Ncal(0,1/k)$ entries and $k \ge 24\eps^{-2}\log N$, then with probability at least $1-1/N$,

$$(1-\eps)\|x_i-x_j\|_2^2 \le \|Ax_i-Ax_j\|_2^2 \le (1+\eps)\|x_i-x_j\|_2^2 \quad\text{for all } i,j .$$

Your proof must establish the single-vector concentration statement, not merely cite it, and must state clearly where the independence and the fixed-in-advance point set are used.