  We present a constructive proof of the existence of an equilibrium in a
competitive economy with sequentially locally non-constant excess demand
functions. And we will show that the existence of such an equilibrium implies
Sperner's lemma. Since the existence of an equilibrium is derived from the
existence an approximate fixed point of uniformly continuous functions, which
is derived from Sperner's lemma, the existence of an equilibrium in a
competitive economy with sequentially locally non-constant excess demand
functions is equivalent to Sperner's lemma.
