Final Exam Statements

Question 0

completeness property

Answer 0

Every nonempty set of real numbers that has an upper-bound also has a supremum in the real numbers

Question 1

triangle inequality

Answer 1

If a and b are real numbers, then |a+b| <= |a| + |b|

Question 2

Monotone Convergence Theorem

Answer 2

A monotone sequence of real numbers is convergent if and only if it is bounded. Further:

(a) If X = (xn) is a bounded increasing sequence, then lim(xn) = sup{xn : n is natural}
(b) If Y = (yn) is a bounded decreasing sequence, then lim(yn) = inf{yn : n is natural}

Question 3

Cauchy Criterion for sequences

Answer 3

A sequence of real numbers is convergent if and only if it is a Cauchy sequence

Question 4

Bolzono-Weierstrauss Theorem

Answer 4

A bounded sequence of real numbers has a convergent subsequence

Question 5

Mean Value Theorem

Answer 5

Suppose that f is continuous on a closed interval I = [a,b], and that f has a derivative in the open interval (a,b). Then there exists at least one point c in (a,b) such that f(b)-f(a) = f’(c)(b-a)

Question 6

Squeeze Theorem for Integrals

Answer 6

Let f be a function from [a,b] to the real numbers. Then f is Riemann integrable if and only if for every epsilon > 0 there exists Riemann integrable functions alpha and omega with alpha(x) <= omega(x) for all x in [a,b] and such that the integral from a to b of omega - alpha is less than epsilon

Question 7

Fundamental Theorem of Calculus Part 2

Answer 7

Let f be a Riemann integrable function on [a,b] and let f be continuous at a point c in [a,b]. Then the indefinite integral is differentiable at c and F’(c) = f(c)

Question 8

Theorem 3.2.2

Answer 8

A convergent sequence of real numbers is bounded

Question 9

Sequential Criterion for Limits

Answer 9

Let f be a function from A to the real numbers and let c be a cluster point of A. Then the following are equivalent.

(i) the limit as x approaches c of f = L
(ii) For every sequence (xn) in A that converges to c such that xn != c for all natural numbers n, the sequence (f(xn)) converges to L

Question 10

Maximum-Minimum Theorem

Answer 10

Let I = [a,b] be a closed bounded interval and let f be a function from I to the real numbers be continuous on I. Then f has an absolute maximum and an absolute minimum on I

Question 11

Theorem 7.2.7

Answer 11

If the function f from [a,b] to the real numbers is continuous on [a,b], then f is Riemann integrable on [a,b]

Question 12

Fundamental Theorem of Calculus Part 1

Answer 12

Suppose f and F are functions from [a,b] to the real numbers such that
(i) F is continuous on [a,b]
(ii) F’(x) = f(x)
(iii) f is Riemann integrable on [a,b]
then the integral from a to b of f = F(b) - F(a)