Chapter 1 - Definitions

Question 1

What is the definition for a sequence converging?

Answer 1

A real sequence a_n converges to the limit L if for each epsilon greater than 0, there exists big N in the positive integers such that for all little n greater than big N, the absolute value of a_n is less than epsilon.

Question 2

What is the definition for an upper bound on a sequence?

Answer 2

A sequence a_n is bounded above if there exists M in the real numbers such that for all little n in the positive integers a_n is smaller than or equal to M

Question 3

What is the definition for a lower bound on a sequence?

Answer 3

A sequence a_n is bounded below if there exists K in the real numbers integers such that for all little n in the positive integers a_n is greater than or equal to K

Question 4

What is it called if a sequence is bounded both above and below?

Answer 4

bounded :)

Question 5

What is the definition for a subsequence?

Answer 5

b_k is a subsequence of a subscript n if there exists a strictly increasing sequence of positive integers n_k such that b subscript k equals a_n_k

Question 6

What is the definition for a sequence increasing?

Answer 6

A sequence a_n is increasing if a_n+1 greater than or equal to a_n for all n in the positive integers

Question 7

What is the definition for a sequence decreasing?

Answer 7

A sequence a_n is decreasing if a_n+1 is less than or equal to a_n for all n in the positive integers

Question 8

What is the definition for a sequence to be monotonic?

Answer 8

A sequence a_n is monotonic if it is increasing or decreasing

Question 9

What is the definition for a sequence to be Cauchy?

Answer 9

A real sequence is Cauchy if for each epsilon greater than 0 there exists a big N in the positive integers such that for all little n,m greater than or equal to big N the absolute value of a_n-a_m is less than epsilon

Question 10

What is the definition of a limit at infinity and what is the shorthand?

Answer 10

Let D be a subset of the real numbers be unbounded above and f goes from d to the real numbers. Then. f has limit L at infinity if for each epsilon greater than 0, there exists K in the real numbers such that for all x in D with x greater than k, the absolute value of f(x)-L is less than epsilon.
The shorthand is lim as x goes to infinity f(x)=L

Question 11

What is the definition for a cluster point?

Answer 11

Let D be a subset of the real numbers. Then a in the real numbers is a cluster point of D, for each epsilon greater than 0, there exists an x in D with the absolute value of x-a be greater than 0 and less than epsilon

Question 12

What is the definition for a limit of a function and what is the shorthand?

Answer 12

Let D be a subset of the real numbers, f goes from D to the real numbers and a in the rea numbers be a cluster point of D. Then, f has a limit L at a if, for each epsilon greater than 0, there exits a delta greater than 0 such that, for all x in D with the absolute value of x-a be greater than 0 and less than delta, the absolute value of f(x)-L is less than epsilon.

Question 13

What is the definition for continuous and discontinuous?

Answer 13

Let f go from D to the real numbers and a be in D. Then f is continuos at a if for all sequence x_n in D such that x_n goes to a, f(x_n) goes to a.
f is continuous if it is continuos at a for all a in D.
If f is not continuos, it is discontinuos.