This is a foundational module aimed at providing the students with the basic concepts and techniques of modern real analysis. The guiding idea will be to start using the powerful tools of analysis, familiar to the students from first-year modules in the context of the real numbers, to vectors (multivariable analysis) and to functions (functional analysis). The notions of convergence and continuity will be reinterpreted in the more general setting of metric spaces. This will provide the language to prove several fundamental results that are in the basic toolkit of a mathematician, like the Picard Theorem on the existence and uniqueness of solutions to first order differential equations with an initial datum, and the implicit function theorem.