# Preliminaries

Several terms and concepts ought be introduced (or briefly reviewed) at the beginning of a real analysis course.

**Notation.**Students will likely encounter several symbols they have never seen before (*e.g.*\(\in\), \(\exists\), \(\forall\)). Such symbols will be interwoven into our conversation, but we lean toward expression in full words when such can be done concisely.**Fields and Ordered Sets.**These are used to provide a rigorous way to discuss the properties of numbers and what manipulations we may do when interacting with them.**Euclidean Spaces.**This widely used extension of real numbers is used in many generalizations of our examples (and applications).**Induction.**The idea of showing a statement holds for all natural numbers is briefly overviewed. This concept is used repeatedly, often as an intermediate step for a proof.**Infimums and Supremums.**A key idea we make great use of regards the existence of a smallest upper bound for a set, called a supremum.