  In this work, we consider a diffuse interface model for tumour growth in the
presence of a nutrient which is consumed by the tumour. The system of equations
consists of a Cahn--Hilliard equation with source terms for the tumour cells
and a reaction-diffusion equation for the nutrient. We introduce a
fully-discrete finite element approximation of the model and prove stability
bounds for the discrete scheme. Moreover, we show that discrete solutions exist
and depend continuously on the initial and boundary data. We then pass to the
limit in the discretization parameters and prove convergence to a
global-in-time weak solution to the model. Under additional assumptions, this
weak solution is unique. Finally, we present some numerical results including
numerical error investigation in one spatial dimension and some long time
simulations in two and three spatial dimensions.
