is a convex subset of and is contractible to each point in by lemma 1.
Let X be and ; let be a contraction such that
For [a] in pi_1(X, x_0), define a homotopy by
A contraction F ensures that H is a homotopy between H( . , 0) = a and H(. , 1) = c, which is a constant loop at x_0. Thus, [a] = [c]. We conclude that pi_1(X, x_0) is a trivial group.