The textbook is designed for a one-semester course at a senior undergraduate level, and appeals to mathematics undergraduates, but also to those who need to learn some mathematical analysis for use in other areas such as engineering, physics, biology, or finance, as well as to mathematics teachers. Topics such as completeness and compactness are approached initially through convergence of sequences in metric space, and the emphasis remains on this approach. However, the alternative topological approach is described in a separate chapter. This makes the book useful as an introduction to more advanced topics such as functional analysis. Nominal divisions of pure and applied mathematics have been merged, leaving enough for students of either inclination to have a feeling for what further developments might look like. Applications have been included such as differential and integral equations, systems of linear algebraic equations, approximation theory, numerical analysis and quantum mechanics.