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## Math 142A Fall 2017 Terminology and Theorems

#### Updated 10/3/17

Note: The following list is a minimal collection of the important terms and theorems for this course. You simply must be familiar with them and how they are used: in the case of theorems, you should be familiar with how they are proved. This basic knowledge is analogous to the vocabulary of a language: it is impossible to speak the language without it.

• Chapter 1
• inductive set
• principle of mathematical induction
• bounded above/below, bounded
• supremum
• infimum
• completeness axiom
• distribution of the integers
• distribution of the rational numbers
• archimedean property
• dense
• the set of rational numbers is dense
• the set of irrational numbers is dense
• triangle inequality
• difference of powers formula
• geometric sum formula
• binomial formula
• Chapter 2
• sequence
• converge
• comparison lemma
• sum, product and quotient properties
• linearity and polynomial properties
• bounded sequence
• sequentially dense set
• closed set
• monotonically increasing/decreasing, monotone
• monotone convergence theorem
• nested interval theorem
• subsequence
• every sequence has a monotone subsequence
• every bounded sequence has a convergent subsequence
• sequentially compact
• sequential compactness theorem
• Chapter 3
• continuous at x0
• continuous
• composition f o g
• image
• maximum value
• maximizer
• minimum value
• minimizer
• extreme value theorem
• intermediate value theorem
• bisection method
• convex set
• uniformly continuous
• ε-δ criterion at a point
• ε-δ criterion on the domain
• monotonically increasing/decreasing
• monotone
• strictly increasing/decreasing
• strictly monotone
• one-to-one
• inverse
• distinct from
• limit point
• limit (of a function)
• Chapter 4
• tangent line
• differentiable
• derivative
• differentiation rules
• sum rule
• product rule
• quotient rule
• derivative of inverse function
• derivative of the composition: chain rule
• Rolle's theorem
• mean value theorem
• identity criterion
• criterion for strict monotonicity
• local maximizer, local minimizer
• Cauchy mean value theorem
• Leibnitz notation

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