From Munkres p.153:
Why does he begin with convex sets? Is it because if we know that convex sets are connected then we can write $L$ as a union of convex sets that have a point in common?
Why doesn't he just assume that $L$ is a disjoint union of open intervals and proceed similarly?
If you start by showing that convex sets are connected, then we immediately know that $L$ is connected, as it is trivially convex.
Then next one shows that intervals and rays are also convex (also pretty simple) and then these are also proved to be connected. It's just that convex sets are a convenient way to reason about connectedness in ordered spaces.
Perhaps Munkres could have been clearer. In a linear continuum, a subspace with more than one member is connected iff it is convex iff it is an interval (where interval here means any type of finite' interval or ray or X itself). His proof, which shows convexity implies connected, only utilizes the single property of convexity, which quickly reduces to the consideration of thefinite' interval [a,b]. If he had tried to directly show intervals are connected, he would have had to consider all the different types of intervals (a lot of cases, a lot of repetition). That would not be as clean a proof.
