Density: point simultaneously close to two given

Question

I see sometimes claims in literature like:

let $(X, \rho)$ be some metric space and $x_k, k \in \mathbb N$ be a dense sequence in it. Suppose we are given two points $x, y$ s. t. $\rho(x, y) < \varepsilon$.

Then, there is an $x_k$ s. t. $\rho(x, x_k) + \rho(y, x_k) < \varepsilon$

How is that possible? I can't see it work unless $x_k$ is exactly on the "line" connecting $x$ and $y$.

Answer 1

Let $x, y$ be fixed. Let $\delta >0$ be small so that $\varepsilon - \rho(x, y) - \delta >0$. Let $x_k$ be such that $\rho(y, x_k) < \min\{\delta, \varepsilon- \rho(x, y) - \delta \}$. Then

$$\rho(x, x_k) \le \rho(x, y) + \rho(y, x_k) < \varepsilon -\delta.$$

Thus we have (using $\rho(y, x_k) <\delta$)

$$\rho(x, x_k) + \rho(y, x_k) < \varepsilon - \delta + \rho(y,x_k) < \epsilon.$$