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Questions + Notes
Lecture 1
1. Why is |sin(nx)| ⇐ n|sin(nx)| ∀ n ∈ N, ∀ x ∈ R?
A) If r = c/d ∈ Q is a rational number and r satisfies the equation c_n*x^2 + c_(n-1)*x^(n-1) + … + c_0 = 0 w/ c_i ∈ Z, c_n ≠ 0, c_0 ≠ 0:
d | c_n, c | c_0 (i.e. factors of constant/factors of leading coefficient is a solution to the equation)
B) Completeness axiom
If S ⊂ R is bounded from above, sup(S) exists in R
If S ⊂ R is bounded from below, inf(S) exists in R
Lecture 2
2. For -S = {-x | x ∈ S}, why is -S bounded above, and why is inf(S) = -sup(-S)?
3. How doe we show that lim a_n = 0 as n → +inf given a_n = sin(n)/n using definition of limit?
A) If max(S) = sup(S), inf(S) = min(S), S is connected:
S is a closed (bounded) interval
B) Checking that sup(S) = M:
Step 1: Check that M is an upper bound of S
Step 2: Check that ∀ α < M, α is not an upper bound of S
C) Archimedian Property:
If a, b > 0, then ∃ n ∈ N s.t. na > b
Lecture 3
4. In the proof of the theorem “All convergent sequences are bounded,” why do we have to consider two different cases 1) n > N, 2) n < N? (n is the index of a sequence, and N > 0 is a number s.t. |a_n - α| < ε ∀ ε > 0)
5. In the proof of lim (a_n*b_n) = (lim a_n)*(lim b_n), what does it mean by “fluctuation of the product a_nb_n 1)”?
Lecture 4
6. Why is lim x^(1/x) = lim e^2) as its root?
10. Why are we able to find the limit of a recursive sequence using the “zig zag trajectory”? (Refer to (B) below)
11. How do we prove that if (S_n) has a subsequence converging to t, ∀ ε > 0, the set A_ε = {n ∈ N | |S_n - t| < ε} is infinite? (Ross)
A) Induction may be useful in proving hypothesis using recursion.
B) Finding the limit of a recursive sequence:
Step 1: Draw graph of y = f(x) and y = x
Step 2: Plot (S_1, S_2) where S_(n+1) = f(S_n)
Step 3: The zig zag trajectory will lead to the limiting point, which is the intersection of y = x and y = f(x)
- Zig zag trajectory: from (S_1, S_2) to y = x, then f(x) corresponding to the x value, then repeat
Step 4: Solve for x = f(x)
C) (S_n) has a subsequence converging to t ∈ R iff ∀ ε > 0, the set A_ε = {n ∈ N | |S_n - t| < ε} is infinite.
Lecture 7
12. How do one construct a monotone subsequence?
A:
Case 1: If there are infinite dominant terms, construct subsequence using the dominant terms.
Case 2: Otherwise, construct a monotone increasing subsequence by picking the subsequent term (n_(k+1)) of this subsequence s.t. S_n_(k+1) >= S_n_k. We know that such n_(k+1) exists because if it doesn't, it means that there are infinite dominant terms, and we can use Case 1 above.
13. What is the “diagonal argument”?
A:
If A_1 = {n | S_n ∈ I_1}, A_2 = {n | S_n ∈ I_2}, …, then
A_1 ⊃ A_2 ⊃ A_3 ⊃ …, and there is a subsequence (S_n_(kk))_k s.t. S_n_k ∈ I_k
A) Every sequence has a monotone subsequence.
B) limsup(S_n) and liminf(S_n) are subsequential limits
C) Closed subset S:
If ∀ convergent sequence in S, the limit also belongs to S
D) If (S_n) is bounded sequence and S is a set of subsequential limit, S is closed.
Lecture 8
14. Why is lim A_N = lim A_n_k as N → +inf on the LHS and k → +inf on the RHS? (A_N = sup(S_n) for n > N)
15. If (t_n_k) is convergent, why is (s_n_k*t_n_k)_k also convergent?
A) Possible convergent subsequence of (s_n*t_n):
Pick a convergent subsequence (t_n_k) in t_n, then (s_n_k*t_n_k)_k is convergent.
B) (s_n) is a sequence of positive numbers.
liminf(s_(n+1)/s_n) ⇐ liminf(s_n)^(1/n) ⇐ limsup(s_n)^(1/n) ⇐ limsup(s_(n+1)/s_n)
C) If a > 0, lim(a^(1/n)) = 1
Lecture 9
16. Why is S = R \ {0} non-complete?
17. Why is lim(s_n) = (s_n)_n if (s_n) is Cauchy?
18. Prove Bolzano-Weierstrass theorem.
A) Complete metric space:
Every Cauchy sequence has a limit in S.
B) R^n is a complete metric space.
C) Every bounded sequence in R^m has a convergent subsequence (Bolzano - Weierstrass).
D) Topology on a set S:
Collection of open subsets.
- S, ∅ are open
- Union of open subsets is open, Finite intersection of open subsets is open
E) Open set for (S, d):
U ⊂ S is open if ∀ p ∈ U, ∃ r > 0, s.t. B_r(p) ⊂ U. Then, U = ∪ B_(r(p))(p). (*p ∈ U)
Lecture 10
