To clarify that I am on the right track in the formal proof, may I run the final version by you?

Proof. (by contradiction). Suppose that

is disconnected. Then, by definition, there exist two disjoint sets,

and

, such that

,

, and

. Now suppose that

and

. Thus there is a line segment between

and

that lies partially in

and partially in

such that

is connected by a finite union of segments of the form

, where the terminal point of one corresponds to the initial point of the next (when applicable). Thus,

forms a polygonal path from

to

. Hence

is polygonally connected and, therefore, connected.