We consider the formation of droplets from a 2-component liquid mixture induced by a large polymer chain that has preferential attraction to one of the components. We assume that the liquid mixture is in a fully miscible state, but far above the critical interaction limit of the two species. We show that the polymer coil acts as a chemical potential trap, which can shift the mixture inside the polymer volume into the partially miscible state and thus triggers the formation of a polymer-bound droplet of the preferred solvent phase, which we denote as polymer-assisted condensation (PAC). We propose a mean-field model which can predict the essential features of PAC including the phase diagram, and we perform molecular dynamics simulations to show that the predicted phase behavior is robust against fluctuation effects. The properties of PAC make it an ideal candidate to understand the formation of biomolecular condensates inside the cell nucleus, such as those formed by the protein heterochromatin 1 (HP1). We propose that such droplets organize the spatial structure of chromatin into hetero- and euchromatin and their predicted stability with respect to the chromatin-HP1-interaction ensures the propagation of epigenetic information through the cell generations.