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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 $i=1,2,..,k-1$.

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    For reference, this is also an exercise in Joshi's Intro to General Topology p. 148.2018-02-10

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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.

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    that's a nice approach, I will try this,thank you.2011-07-05
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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.