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Math 142A Fall 2022
Terminology and Theorems
Updated 2/8/23
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.
- Section 1.1 (review)
- ℕ, the set of positive integers
- Peano Axioms
- mathematical induction
- Section 1.2
- ℚ, the set of rational numbers
- algebraic number
- k divides m
- rational zeros theorem
- monic polynomial
- Section 1.3
- ℝ, the set of real numbers
- associative laws
- commutative laws
- field
- transitive law
- ordered field
- the absolute value of a
- the distance between a and b
- the triangle inequality
- Section 1.4
- maximum, minimum
- upper bound, lower bound
- bounded above, bounded below
- bounded
- supremum, infimum
- completeness axiom for ℝ
- Archimedean property of ℝ
- density (denseness) of ℚ in ℝ
- Section 1.5
- +∞, -∞
- unbounded closed intervals
- unbounded open intervals
- sup S = +∞, inf S = -∞
- Section 2.7
- (sn) converges to s
- lim sn = s
- sn → s
- diverges
- Section 2.8
- Section 2.9
- Convergent sequences are bounded.
- sn → s ⇒ c sn → c s
- sn → s and tn → t ⇒ sn + tn → s + t
- sn → s and tn → t ⇒ sn tn → s t
- n1/n → 1
- definition of lim sn = +∞ , sn → +∞
- definition of lim sn = -∞ , sn → -∞
- (sn) has a limit; lim sn exists
- sn → +∞ and tn → a with a > 0 or a = +∞ ⇒ sn tn → +∞
- for sn > 0: sn → +∞ ⇔ 1/sn → 0
- Section 2.10
- increasing sequence
- decreasing sequence
- monotone (monotonic) sequence
- bounded monotone sequence
- Bounded monotone sequences converge.
- unbounded monotone sequence
- lim sup sn
- lim inf sn
- Cauchy sequence
- A sequence is convergent if and only if it is Cauchy.
- Section 2.11
- subsequence
- if sn converges to s, then every subsequence snk converges to s.
- every sequence has a monotonic subsequence
- Bolzano-Weierstrass Theorem
- Section 2.12
- Section 2.14
- partial sum
- converge, convergent
- diverge, divergent
- absolutely convergent
- geometric series
- Cauchy criterion
- comparison test
- absolutely convergent series are convegent
- ratio test
- root test
- Section 2.15
- alternating series test
- integral test
- p-test
- Section 3.17
- domain of a function f, dom(f)
- continuous at x0
- continuous on S ⊆ dom(f)
- continuous
- Section 3.18
- extreme value theorem (18.1)
- intermediate value theorem
- Section 3.19
- uniformly continuous on a set S
- f continuous on [a,b] implies f uniformly continuous on [a,b] (19.2)
- Section 3.20
- limits of functions
- limit as x tends to a along S
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