Analysis I Flashcards
Give the axioms of a field
Addition is: unique, commutative & associative and there exists an additive identity & all real numbers have an additive inverse.
Multiplication is: unique, commutative & associative and there exists a multiplicative identity & all real numbers have a multiplicative inverse.
Multiplication distributes over addition
0 ≠ 1
Give the axioms of ordering on the real numbers
If a and b are positive, a + b is positive
If a and b are positive, ab is positive
Either a is positive, a=0 or -a is positive
Define a > b
a − b ∈ Positive numbers
Define a ≥ b
b - a is either positive or 0
Define an upper bound
b is an upper bound of S if s ≤ b for all s in S
Define a lower bound
b is a lower bound of S if s ≥ b for all s in S
Define what it means for a set to be bounded
A set, S is bounded if there exists an upper and a lower bound
Define a supremum
Let S be a set. Then α is the supremum of S if:
s ≤ α for all s ∈ S
s ≤ b for all s ∈ S => α ≤ b
Give the completeness axiom of the real numbers
Let S be a non-empty subset of R which is bounded above. Then sup S
exists.
Define an infimum
Let S be a set. Then α is the infimum of S if:
α ≤ s for all s ∈ S
b ≤ s for all s ∈ S => b ≤ α
Define a maximum of a set
For a non empty set, a maximum element max(S) is an element in S such that every element in S is ≤ max(S)
Define a minimum of a set
For a non empty set, a minimum element min(S) is an element in S such that every element in S is ≥ min(S)
Define |a|
|a| = a if a > 0
|a| = 0 if a = 0
|a| = -a if a < 0
Define countably infinitie
A is countably infinite if there exists a bijection between A and the natural numbers
Define countable
A is countable if there exists an injection from A to the natural numbers
Give the approximation property
Let S be non-empty and bounded above (so sup S exists). Then, given ε > 0, there exists s in S such that: sup S − ε < s < sup S
Give the density properties
Given a, b ∈ R with a < b there exists x ∈ Q such that a < x < b.
Given a, b ∈ R with a < b there exists y ∈ R \ Q such that a < y < b.
Define a real sequence
A map α:N->R (ie. a sequence is ordered)
Define convergence
a_n converges to r if ∀ ε > 0, ∃N in the natural numbers such that |a_n - r|< ε for all n ≥ N
Define divergence
a_n is divergent if there is no r which a_n converges to.
Define a tail
b_n is a tail of a_n if b_n = a_(n+k) for some k in the natural numbers
Give the Sandwich lemma
Let (a_n), (b_n) and (c_n) be real sequences with a_n ≤ b_n ≤ c_n for all n ≥ 1. If a_n → L and c_n → L as n → ∞, then b_n → 0 as n → ∞
Define a bounded sequence
If a_n is a sequence, we say that it is bounded if the set {an : n ≥ 1} is bounded.
Define a_n → ∞
∀M ∈ R ∃N in the natural numbers such that ∀n ≥ N, a_n > M
Define convergence for a complex sequence
z_n -> L means that ∀ε > 0, ∃N in the natural numbers such that ∀n ≥ N, |zn − L| < ε.
Define a subsequence
If a_n for n ≥ 0 (natural numbers) is a sequence, b_r, where r ≥ 0 (natural numbers) is a subsequence of a_n provided that there is a function f : n → r such that f is strictly increasing.
Define a_n = O(b_n)
If a_n and b_n are sequences, a_n = O(b_n) as n → ∞, means that there is a constant C ∈ R > 0 and N such that if n ≥ N then |an| ≤ C|bn|.
Define a_n = o(b_n)
If b_n ≠ 0 for all n (or all sufficiently large n), then a_n = o(b_n) as n → ∞ means that a_n b_n → 0 as n → ∞.
Define monotone increasing
a_n is monotone increasing if a_n ≤ a_(n+1) for all n
Define strictly increasing
a_n is strictly increasing if a_n < a_(n+1) for all n
Define a Cauchy sequence
∀ε > 0, ∃N in the natural numbers such that |a_n - a_m| < ε for all n,m ≥ N
Define a rearrangement of a series
Rearranging a series means that we choose a bijection: N -> N that gives a new series
Define the exponential function
Sum to infinity of z^k/k!
Define the sine function
Sum to infinity of (-1)^k * z^(2k)/(2k+1)!
Define the cosine function
Sum to infinity of (-1)^k * z^(2k)/(2k)!
Define the radius of convegence
R = sup{|z|:sum of (c_k)(z^k) converges}
Define a limit point
p is a limit point if ∀ε > 0, ∃s ≠ p such that 0 < |s-p| < ε
