TITLE: Connectedness problem: sequences of points with distances at most $\varepsilon$
QUESTION [9 upvotes]: Here's problem 6.1.D (a), page $359$ from Engelking's book, stuck with it for a while.
Verify that if a space $X$ with the topology induced by a metric $p$ is connected, then for every pair $x,y$ of points of $X$ and any $\varepsilon >0$ there exists a finite sequence $x_{1},x_{2},..,x_{k}$ of points of $X$ such that $x_{1}=x$, $x_{k}=y$ and $p(x_{i},x_{i+1})<\varepsilon$ for $i=1,2,..,k-1$.

REPLY [2 votes]: 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.