Sequence

Question 1

bounded from above

Answer 1

A sequence {an} is called bounded from above, if there
exists K ∈ R such that
∀n ∈ Z^+ : an ≤ K.

Question 2

increasing monotonically

Answer 2

if an ≤ an+1 for every positive integer n

Question 3

strictly increasing

Answer 3

if an < an+1 for every positive integer n

Question 4

decreasing monotonically

Answer 4

if an ≥ an+1 for every positive integer n

Question 5

strictly decreasing

Answer 5

if an > an+1 for every positive integer n

Question 6

converges I.

Answer 6

A sequence {an} converges to a real number A, if ∀ε > 0 there exists a positive integer nε such that |an − A| < ε whenever
n > nε.

Question 7

converges II.

Answer 7

A sequence {an} converges, to a real number A, if an
arbitrary neighbourhood of A contains the members of the sequence
with finitely many exception

Question 8

Limit

Answer 8

The limit of a convergent sequence is unique.

Question 9

The necessary condition of convergence

Answer 9

If a sequence {an} converges, then its bounded

Question 10

Subsequences thorem

Answer 10

If the original sequence converges, then its subsequence is
also converges, and its limit is the same, as the limit of the original
sequence.

Question 11

Subsequences Definition

Answer 11

A subsequence is a sequence that can be derived from
another sequence by deleting some elements without changing the
order of the remaining elements.

Question 12

Subsequences

Answer 12

If every subsequence derived from {an} converges, then {an}
is convergent as well

Question 13

Subsequences

Answer 13

If the sequence {an} converges, then adding finitely many
elements to it, the new sequence is also convergent with the same
limit.

Question 14

The sufficient condition of convergence

Answer 14

If the sequence {an} is monotonic and bounded, then it
converges. If its increasing monotonically, then its limit is the least
upper bound of the sequence. Similarly, if its decreasing monotonically,
then the limit is the greatest lower bound of the sequence.

Question 15

Bolzano-Weierstrass theorem

Answer 15

Each sequence has a monotonic subsequence.

Question 16

Bolzano-Weierstrass theorem

Answer 16

(Bolzano-Weierstrass) Each bounded sequence has a
convergent subsequence.

Question 17

The Cauchy convergence test

Answer 17

m (The Cauchy convergence test)
The sequence {an} converges if and only if for every positive real
number ε there exists a positive integer nε, such that n, m > nε implies
|an − am| < ε.

Question 18

Operations and limits

Answer 18

If the sequence {an} converges to the real number A, then
∀c ∈ R the sequence {can} converges to cA.

Question 19

Operations and limits

Answer 19

If the sequences {an} and {bn} converge, lim(n→∞)an = A and
lim(n→∞)bn = B, then their sum {an + bn} converges as well and
lim(n→∞)(an + bn) = A + B.

Question 20

Sandwich theorem (squeeze theorem)

Answer 20

If the sequence {an} and {cn} converge and ∀n ∈ Z+
an ≤ bn ≤ cn, furthermore lim(n→∞)an = lim(n→∞)cn = A, then {bn} converges
to A as well.