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.