Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 20F / b / Solution

One form of the Baire category theorem is: a nonempty complete metric space cannot be a countable union of closed nowhere dense subsets. Equivalently, a countable intersection of open dense subsets is dense. A Banach space is a complete metric space for its norm metric, so this form applies directly to the next part.