Real Analysis I

Real Analysis I generalizes the calculus notions of distance, limit, and continuity to abstract metric spaces, so you stop reasoning about the real line specifically and start reasoning about any space where "closeness" makes sense. Expect weekly problem sets and written exams built around proofs: showing a space is complete, characterizing compact or connected subsets, and using the contraction mapping theorem to pin down solutions of integral equations. It's the bridge between MATH 225-style intro analysis and the functional analysis, topology, and ODE/PDE courses that follow, where treating function spaces as metric spaces is the whole game.