Prove that every convex region is simply connected

Prove that every convex region is simply connected

Note that a region X is convex iff for every x and y in X, the line segment connecting the two is contained entirely in X.

First, we state that X must be path connected since for any x,y in X, the line segment connecting them constitutes a path.

Now, to show that X is simply connected, we consider any loop at x contained in X. Now, every point in the loop has a linear path connecting it to x. Thus, we may form a straight-line homotopy (see source) between a loop and the point x. Since every loop is path nulhomotopic, we know that X is simply connected.

QED