A result about connectedness and closed set.

Show that if $F$ is a closed and connected subset of a metric space $X$ then for every pair of points $a,b\in F$ and each $r>0$ there are points $z_0,z_1,\ldots,z_n$ in $F$ with $z_0=a$, $z_n=b$ and $d(z_{k-1},z_k)<r$ for $0<k<n+1$. Is the hypothesis that $F$ be closed needed? If $F$ is a set which satisfies this property then $F$ is not necessarily connected, even if $F$ is closed. Give an example to illustrate this.

I really don't understand how I will start. Please someone give me some hints.

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First part: Consider a point $a\in F$ and $r>0$ and let $B$ be the set of points $b$ for which such a sequence exists. Show that $B$ is nonempty, open, closed (as subset of $F$).
Third part: Consider $F=X=\mathbb R\setminus\{0\}$.