sequences and series

Question 1

A number L ∈ R is a limit of the se- quence {an}∞n=1 if for every ε > 0 there exists an N(ε) such that for all n > N(ε), we have
|an − L| < ε.

Answer 1

A number L ∈ R is a limit of the se- quence {an}∞n=1 if for every ε > 0 there exists an N(ε) such that for all n > N(ε), we have
|an − L| < ε.

an infinite sequence converges to L if for any small interval [L - epsilon, L + epsilon], the sequences converges for all elements apart from a finite amount of elements.

Question 2

a sequence being bounded does not mean it has a limit

Answer 2

if a sequence is unbounded, it diverges

Question 3

if a sequence has a limit, that implies it’s bounded

Answer 3

if a sequence has a limit, that implies it’s bounded

Question 4

a sequence that is increasing or decreasing is monotonic

Question 5

we say a sequence is strictly increasing if

Answer 5

an < an+1.

Question 6

we say a sequence is strictly decreasing if

Answer 6

an > an+1.

Question 7

We say a sequence is bounded above if there is a number M, called an upper bound, such that for all n ∈ N
an ≤ M.
It is bounded below if there is a number m, called a
lower bound, such that for all n ∈ N an ≥ m.

Question 8

every bounded monotonic sequence converges