Existence of shadow prices in finite probability spaces

A shadow price is a process $\tilde{S}$ lying within the bid/ask prices $\underline{S},\overline{S}$ of a market with proportional transaction costs, such that maximizing expected utility from consumption in the frictionless market with price process $\tilde{S}$ leads to the same maximal utility as in the original market with transaction costs. For finite probability spaces, this note provides an elementary proof for the existence of such a shadow price.