Foundations Series

Question 1

Monotone Convergence Series

Answer 1

Every Bounded Sequence of Reals is convergent.

Question 2

Real Bolzano Weirestrass

Answer 2

Every bounded real sequence has a convergent subsequence.

Question 3

Complex Bolzano Weirestrass

Answer 3

For every bounded complex sequence there exists a convergent subsequence.

Question 4

Lemma : Convergency and Subsequences

Answer 4

If a_n is convergent, all its subsequence are convergent with the same limit.

Question 5

Def 6.1 : existence of subsequence b_j

Answer 5

a_n is a sequence. b_j is called a subsequence if there exists strictly increasing sequence n_j of natural number such that b_j = a_n_j

Question 6

Lemma 5.17 n greater than X

Answer 6

for all x in real number, there exists n in natural number such that n is greater than X.

Question 7

Lemma 6.9 Convergent Bounded

Answer 7

Every convergent sequence is bounded

Question 8

Def 6.7 real sequence bounded above/below

Answer 8

A real sequence is bounded above/ below only if the set S = {a_n | n E N } is bounded above / below.
Infinum and Supremum is inherited.

Question 9

Lemma 6.22 Complex Convergent

Answer 9

Let Z_n be a complex sequence. It is convergent if and only if its real and imaginary sequences are convergent.

Question 10

Power Set

Answer 10

Set of all possible subsets of a given set, including the empty set and itself.

Question 11

Proposition 6.18 a_n > c …

Answer 11

a_n is a convergent real sequence.
if a_n > c then :
lim a_n ≥ c
(n to infinity)

Question 12

Cauchy Sequence

Answer 12

For all epsilon greater than 0 there exits N in natural number such that all n,m both greater than N ; |a_n - a_m| > epsilon

Question 13

Convergent Series

Answer 13

For all epsilon greater than 0 there exits N in natural number such that all n greater than N ; |a_n - a| > epsilon

Question 14

Algebra of Limits

Answer 14

The Algebra of Limits provides rules for finding the limit of a function that is an algebraic combination (sum, difference, product, quotient, power) of simpler functions.