We generalize classical results on the existence of optimal portfolios in discrete time frictionless market models to models with capital gains taxes. We consider the realistic but mathematically challenging rule that losses do not trigger negative taxes but can only be offset against potential gains in the future. Central to the analysis is a well-known phenomenon from arbitrage-free markets with proportional transaction costs that does not exist in arbitrage-free frictionless markets: an investment in specific quantities of stocks that is completely riskless but may provide an advantage over holding money in the bank account. As a result of this phenomenon, on an infinite probability space, no-arbitrage does not imply that the set of attainable terminal wealth is closed in probability. We show closedness under the slightly stronger {\em no unbounded non-substitutable investment with bounded risk} condition. As a by-product, we provide a proof that in discrete time frictionless models with short-selling constraints, no-arbitrage implies that the set of attainable terminal wealth is closed in probability -- even if there are redundant stocks.