Understanding the proof of the Lebesgue Covering Lemma.

Question

Here is the statement of the lemma and its proof (from Royden "Real Analysis 4th edition on pg.201 & 202").

But I do not understand how the author applied the triangle inequality, specifically I do not understand the last paragraph, why we were in need of an index $k$? why we take $1/n_{k} < (r_{0}/2).$ Could anyone explain this to me, please?

Answer 1

For $x\in B(x_{n_k}, \frac{1}{n_k})$ $$d(x, x_0)\le d(x, x_{n_k} ) + d(x_{n_k}, x_0)\le \frac{1}{n_k} + \frac{r_0}{2} < r_0$$

That's the application of the triangle inequality.

So $x\in B(x_0, r_0)$, and consequently, $ B(x_{n_k}, \frac{1}{n_k}) \subset B(x_0, r_0)$