Equivalence of Definitions of Connected Set (Complex Analysis)

$(1)$ implies $(2)$
Let $D$ be a connected set by definition 1.

Then by definition:
 * Every pair of points in $D$ can be joined by a staircase contour.

But a staircase contour is a polygonal path all of whose points are in $D$.

Thus $D$ is a connected set by definition 2.

$(2)$ implies $(1)$
Let $D$ be a connected set by definition 2.

Then by definition:
 * Every pair of points in $D$ can be joined by a polygonal path $P$ all points of which are in $D$.

Thus $D$ is a connected set by definition 1.