Question 2

Question 1

Definition of convergence:

Answer 1

A sequence (an) converges to the limit a∈R if ∀ε>0 ∃ N∈Nat s.t. n>N then |an-a|< ε. an → a as n→ ∞. We say that the sequence (an) converges to a. If a=0 we call (an) a null sequence.

Question 2

Definition of divergence:

Answer 2

(i) A sequence is divergent if it is not convergent.
(ii) A sequence is divergent to ∞ if ∀M>0 ∃ N∈Nat s.t. an>M ∀ n>N. We write an→∞ as n→∞.
(iii) A sequence is divergent to -∞ if ∀M>0 ∃ N∈Nat s.t. an>-M ∀ n>N. We write an→-∞ as n→∞.

Question 3

What is every convergent sequence?

Answer 3

Bounded

Question 4

Definition of a Cauchy Sequence:

Answer 4

We say that a sequence (an) of real numbers is a Cauchy sequence if ∀ε>0 ∃ N∈Nat s.t. n,m>N then |an-am|< ε.

Question 5

Triangle Inequality:

Answer 5

/a+b/ ≤ /a/ +/b/
/a+b/ ≥ //a/-/b// (lower triangle inequality)