Definitions

Question 1

Least Upper Bound Axiom

Answer 1

Any non-empty set of numbers which is bounded above must have a least upper bound

Question 2

Archimedian Property of Real Numbers

Answer 2

For any real number r, there exists n∈ℕ such that n > r

Question 3

Convergence of a Sequence

Answer 3

A sequence converges to a limit l∈ℝ if ∃ N∈ℕ such that |xn-l|<ε ∀n≥N

Question 4

Sandwich Rule

Answer 4

Suppose that an ≤ bn ≤ cn, where an → l and cn → l. Then bn → l.

Question 5

Convergence to Infinity of a Sequence

Answer 5

A sequence (an) converges to ∞ as n → ∞ if for every R∈ℝ there exists N such that an > R for every n ≥ N

Question 6

Bolzano-Weierstrass Theorem

Answer 6

Any bounded sequence of real numbers
contains a convergent subsequence

Question 7

Cauchy Sequences

Answer 7

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

Question 8

Convergence of Series

Question 9

Ratio Test for Sequences

Answer 9

Suppose that (an) is a sequence of positive terms, and that limn→∞ (an+1/an)=r
If r < 1 then an → 0, and if r > 1 then an → ∞.

Question 10

Comparison Test for Series

Answer 10

Suppose that 0 ≤ an ≤ bn for every n. Then,
if Σbj converges then Σaj converges;
if Σaj diverges then Σbj diverges

Question 11

Ratio Test for Series

Question 12

Root Test for Series

Question 13

Integral Test for Series

Question 14

Alternating Series Test

Question 15

Continuity at a Point

Question 16

Sequential Continuity

Question 17

Intermediate Value Theorem

Question 18

Extreme Value Theorem

Question 19

Uniform Continuity

Question 20

Existence of an Inverse