Definitions

Question 1

Upper bound

Answer 1

for a set S is a real number such that s ≤ r for each s∈S

Question 2

Lower bound

Answer 2

for a set S is a real number such that s ≥ r for each s∈S

Question 3

Least upper bound (completeness) axiom

Answer 3

Any non-empty set that is bounded above has a least upper bound

Question 4

Greatest lower bound axiom

Answer 4

Any non-empty set that is bounded below has a greatest lower bound

Question 5

Archimedean property of the real numbers

Answer 5

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

Question 6

Triangle inequality

Answer 6

|a+b| ≤ |a| + |b|

Question 7

Reverse Triangle inequality

Answer 7

|a - b| ≥ ||a|-|b||

Question 8

Sequence

Answer 8

An infinite ordered list of real numbers

Question 9

Convergent sequence

Answer 9

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

Question 10

Shift rule

Answer 10

For any fixed natural number k, aₙ converges to l if and only if aₙ₊ₖ converges to l

Question 11

Sandwich rule

Answer 11

Suppose that aₙ≤ bₙ≤ cₙ where aₙ and cₙ converge to l, then bₙ converges to l

Question 12

Comparison test (sequences)

Answer 12

aₙ≥bₙ and bₙ tends to infinity, then aₙ tends to infinity

Question 13

Ratio Test (Sequences)

Answer 13

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 → ∞.

Question 14

Bolzano-Weierstrass Theorem

Answer 14

Any bounded sequence of real numbers contains a convergent subsequence

Question 15

Cauchy Sequence

Answer 15

A sequence is Cauchy if for each ε>0 there exists an N such that |aₙ - aₘ|<ε for all m,n≥N

Question 16

General Principle of convergence

Answer 16

A sequence converges if and only if it is Cauchy

Question 17

Vanishing tails

Answer 17

If ∑aₙ converges then ∑ⱼ₌ₙ aⱼ converges to 0

Question 18

Comparison test (Series)

Answer 18

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

Question 19

Absolute convergence

Answer 19

∑aₙ converges absolutely if ∑|aₙ| <∞

Question 20

Comparison Test (Series V2)

Answer 20

Suppose that |aₙ|≤ bₙ for every n, and ∑bₙ< ∞. Then ∑aₙ converges

Question 21

Ratio Test (Series)

Answer 21

If aₙ≠0 and |aₙ₊₁/aₙ| converges to r then if:
r<1, ∑aₙ converges absolutely
r>1, ∑aₙ does not converge

Question 22

Cauchy’s Root test

Answer 22

Suppose that |aₙ|^(1/n) converges to r then if:
r<1, ∑aₙ converges absolutely
r>1, ∑aₙ does not converge

Question 22

Integral Test

Answer 22

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

Question 23

Alternating Series Test

Answer 23

Suppose that aₙ≥0 with aₙ₊₁≤ aₙ and aₙ converges to 0. Then ∑ₙ₌₁(-1)ⁿ⁺¹aₙ converges

Question 24

Continuous function

Answer 24

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

Question 25

δ-neighbourhood

Answer 25

x∈E with |x-c|<δ

Question 26

Sequential Continuity

Answer 26

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)

Question 27

Intermediate value theorem

Answer 27

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

Question 28

Extreme value theorem

Answer 28

If f:[a,b]→ℝ is continuous then f is bounded and attains it’s bounds

Question 29

Uniform Continuity

Answer 29

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

Question 30

Inverse function theorem

Answer 30

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

Question 31

Interval

Answer 31

A non-empty subset I of ℝ is an interval if whenever x,y∈I and x<y then [x,y]∈I

Question 32

(Strictly) Increasing

Answer 32

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)

Question 33

(Strictly) Decreasing

Answer 33

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)