How to show that if $M$ (a metric space) is compact and chain-connected then it is connected ?
Definition of $\varepsilon$-chainable :
$(X,d)$ is $\varepsilon$-chainable if given any two points $a,b\in X$, there exists a finite set $\{x_0,x_1,...,x_n\}$ of points in $X$ with $x_0=a$, $x_n=b$ and $d(x_{k−1},x_k)<\varepsilon$, for all $0<k\leq n$.
A metric space is chain-connected if it is $\varepsilon$-chainable for all $\varepsilon$.