Math 142A Winter 2024
Terminology and Theorems

Updated 1/7/24

• 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
  • (s_n) converges to s
  • lim s_n = s
  • s_n → s
  • diverges
• Section 2.8
  • formal proofs
• Section 2.9
  • Convergent sequences are bounded.
  • s_n → s   ⇒   c s_n → c s
  • s_n → s   and   t_n → t   ⇒   s_n + t_n → s + t
  • s_n → s   and   t_n → t   ⇒   s_n t_n → s t
  • n^1/n → 1
  • definition of   lim s_n = +∞   ,   s_n → +∞
  • definition of   lim s_n = -∞   ,   s_n → -∞
  • (s_n) has a limit;   lim s_n exists
  • s_n → +∞   and   t_n → a   with   a > 0   or   a = +∞   ⇒   s_n t_n → +∞
  • for s_n > 0:   s_n → +∞   ⇔   1/s_n → 0
• Section 2.10
  • increasing sequence
  • decreasing sequence
  • monotone (monotonic) sequence
  • bounded monotone sequence
  • Bounded monotone sequences converge.
  • unbounded monotone sequence
  • lim sup s_n
  • lim inf s_n
  • Cauchy sequence
  • A sequence is convergent if and only if it is Cauchy.
• Section 2.11
  • subsequence
  • if s_n converges to s, then every subsequence s_nk converges to s.
  • every sequence has a monotonic subsequence
  • Bolzano-Weierstrass Theorem
• Section 2.12
  • lim inf s_n
  • lim sup s_n
• 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

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