Definitions Flashcards
Upper bound
for a set S is a real number such that s ≤ r for each s∈S
Lower bound
for a set S is a real number such that s ≥ r for each s∈S
Least upper bound (completeness) axiom
Any non-empty set that is bounded above has a least upper bound
Greatest lower bound axiom
Any non-empty set that is bounded below has a greatest lower bound
Archimedean property of the real numbers
For any real number r, there exists a natural number n such that n>r (the natural numbers are not bounded above by the real numbers
Triangle inequality
|a+b| ≤ |a| + |b|
Reverse Triangle inequality
|a - b| ≥ ||a|-|b||
Sequence
An infinite ordered list of real numbers
Convergent sequence
A sequence converges to a limit l∈ℝ if for every ε>0 there exists N∈ℕ such that |aₙ-l|<ε for every n≥N
Shift rule
For any fixed natural number k, aₙ converges to l if and only if aₙ₊ₖ converges to l
Sandwich rule
Suppose that aₙ≤ bₙ≤ cₙ where aₙ and cₙ converge to l, then bₙ converges to l
Comparison test (sequences)
aₙ≥bₙ and bₙ tends to infinity, then aₙ tends to infinity
Ratio Test (Sequences)
Suppose that (aₙ) is a sequence of positive terms,
and that the limit of aₙ₊₁/aₙ = r
If r < 1 then aₙ → 0, and if r > 1 then aₙ → ∞.
Bolzano-Weierstrass Theorem
Any bounded sequence of real numbers contains a convergent subsequence
Cauchy Sequence
A sequence is Cauchy if for each ε>0 there exists an N such that |aₙ - aₘ|<ε for all m,n≥N
General Principle of convergence
A sequence converges if and only if it is Cauchy
Vanishing tails
If ∑aₙ converges then ∑ⱼ₌ₙ aⱼ converges to 0
Comparison test (Series)
Suppose that 0≤ aₙ≤ bₙ for every n. Then:
if ∑ⱼ₌₁ bⱼ converges then ∑ⱼ₌₁ aⱼ converges
if ∑ⱼ₌₁ aⱼ diverges then ∑ⱼ₌₁ bⱼ diverges
Absolute convergence
∑aₙ converges absolutely if ∑|aₙ| <∞
Comparison Test (Series V2)
Suppose that |aₙ|≤ bₙ for every n, and ∑bₙ< ∞. Then ∑aₙ converges
Ratio Test (Series)
If aₙ≠0 and |aₙ₊₁/aₙ| converges to r then if:
r<1, ∑aₙ converges absolutely
r>1, ∑aₙ does not converge
Cauchy’s Root test
Suppose that |aₙ|^(1/n) converges to r then if:
r<1, ∑aₙ converges absolutely
r>1, ∑aₙ does not converge
Integral Test
Suppose that f:[1,∞) → [0,∞) is a non-negative decreasing function.
If ∫₁ⁿf(x) dx is bounded then ∑ₙ₌₁f(n) converges
If ∫₁ⁿf(x) dx is unbounded then ∑ₙ₌₁f(n) diverges
Alternating Series Test
Suppose that aₙ≥0 with aₙ₊₁≤ aₙ and aₙ converges to 0. Then ∑ₙ₌₁(-1)ⁿ⁺¹aₙ converges
Continuous function
A function f:E →ℝ is continuous at c∈E if for any ε>0 there exists a δ>0 such that x∈E and |x-c|<δ => |f(x) - f(c)|<ε
δ-neighbourhood
x∈E with |x-c|<δ
Sequential Continuity
A function f:E →ℝ is sequentially continuous at c∈E if whenever (xₙ)∈E and xₙ converges to c then f(xₙ) converges to f(c)
Intermediate value theorem
Suppose that f is cts on [a,b] and that f(a)<f(b). Then for any g with f(a)<g<f(b) there exists c∈(a,b) such that f(c)=g
Extreme value theorem
If f:[a,b]→ℝ is continuous then f is bounded and attains it’s bounds
Uniform Continuity
A function f:E →ℝ is uniformly continuous on E if for any ε>0 there exists a δ>0 such that x∈E and |x-y|<δ => |f(x) - f(y)|<ε for any x,y∈E
Inverse function theorem
Let I be an interval and suppose that f:I →ℝ is continuous and strictly monotonic. Then J=f(I) is an interval and f⁻¹:J →I is continuous and strictly monotonic
Interval
A non-empty subset I of ℝ is an interval if whenever x,y∈I and x<y then [x,y]∈I
(Strictly) Increasing
Let E⊂ℝ. A function f:E →ℝ is said to be increasing (on E) if x≥y => f(x)≥f(y) and strictly increasing if x>y => f(x)>f(y)
(Strictly) Decreasing
Let E⊂ℝ. A function f:E →ℝ is said to be decreasing (on E) if x≤y => f(y)≤f(x) and strictly decreasing if x<y => f(y)<f(x)
