Connectedness problem: sequences of points with distances at most $\varepsilon$

Hint. Fix a point $x\in X$. Now put $$Q = \{y\in X| \exists x_1, x_2, \cdots ,x_n\in X \hbox{ with } p(x_k,x_{k+1}) < \epsilon\}.$$ Show that $Q$ is both open and closed in $X$. Your result will follow right away.

This is a typical application of the chain lemma from my answer here. Use the open cover by balls of radius $\varepsilon$, and use the centres plus intersection points.