In k-NN, how does the condition $k(N)/N \to 0$ ensure that all the k nearest neighbors are close to a given test point $\mathbf{x}$?

Consider the k-NN algorithm and let $k(N)$ be the choice of k as a function of N (data points). For $N \to \infty$ , if $k(N) \to \infty$ and $k(N)/N \to 0$ , then k-NN converges to optimal classifier. Question: How does the condition $k(N)/N \to \infty$ ensure that all the k nearest neighbors are close to a given test point $\mathbf{x}$ ? For content, please see: https://amlbook.com/eChapters/6-Oct2022-readeronly.pdf Theorem 6.2 Password: Paraskavedekatriaphobia