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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 n > N and 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
19. Prove that the closure of E is the union of E and E'.
20. In the proof that K = {1, 1/2, 1/3, …} is not compact, how is one able to conclude that there is no proper subcover of {B_S_n(1/n)}_n?
21. How can one conclude that E ⊂ G_a_1 ∪ … ∪ G_a_N from K ⊂ E^c ∪ G_a_1 ∪ … ∪ G_a_N?
A) Closed set for (S, d):
E ⊂ S is closed iff E^c is open
B) Intersection of closed subsets is closed, Finite union of closed sets is closed
C) Closure for E ⊂ S:
Intersection of closed subsets of S that are supersets of E
D) Interior of E:
E^o = {p ∈ E | ∃ δ > 0, B_δ(p) ⊂ E}
E) Boundary:
(Closure of E) \ (Interior of E)
F) Limit point:
E ⊂ S. A point p ∈ S is a limit point of E if ∀ ε > 0, ∃ q ∈ E, q ≠ p s.t. d(p, q) < ε
E' is the set of limit points of E
G) (Closure of E) = E ∪ E'
H) Compact subset:
K ⊂ S is compact if for any open cover of K, we can find a finite subcover.
I) Open cover:
E ⊂ S. An open cover of E is a collection of open sets s.t. the union of the open sets is a superset of E
J) K ⊂ R^n. K is compact iff K is closed and bounded.
K) Showing K is closed:
Show ∀ y ∈ K^c, ∃ δ > 0 s.t. B_δ(y) ∩ K = ∅
Lecture 11
A) If ∑(a_n) converges, then lim a_n = 0
B) Absolute convergence:
Sum of the absolute value of terms converges
C) Root test: α = limsup(|a_n|^(1/n))
Case 1: α > 1, then the series diverges
Case 2: α < 1, then the series converges absolutely
Case 3: α = 1, then the series could converge or diverge
D) Ratio test:
Case 1: limsup|a_(n+1)/a_n| > 1, then the series diverges
Case 2: limsup|a_(n+1)/a_n| ⇐ 1, then the series converges absolutely
E) Alternating Series:
Sum of (-1)^(n+1)*a_n, a_n > 0
If a_1 >= a_2 >= a_3 >= …, a_n >= 0, lim(a_n) = 0, then the series converges.
F) Integral Test:
Draw and see if the integral is greater than or less than the series (be mindful of the bounds as well)
Lecture 12
22. How does the notion that f(B_δ(p)) ⊂ B_ε(f(p)) ⊂ V conclude that B_δ(p) ⊂ f^-1(V)? And how does this conclusion lead to the fact that f^-1(V) is open?
23. Show that x: R → R is continuous.
A) A function f: X → Y is continuous at p ∈ X, if ∀ ε > 0, ∃ δ > 0 s.t. ∀ x ∈ X, with d_x(x, p) < δ, d_y(f(x), f(p)) < ε
B) A function f: X → Y is continuous iff ∀ V ⊂ Y open, f^-1(V) is open
C) Limit of a function:
E ⊂ X, f: E → Y, p is a limit point of E. lim f(x) = q as x → p if ∃ q ∈ Y s.t. ∀ ε > 0, ∃ δ > 0 s.t. f((punctured ball with center at p and radius δ) ∩ E) ⊂ B_ε(q)
D) ex) E = (0, 1) → E' = [0, 1]
ex) E = {1/n, n is a positive integer} → E' = {0}
E) lim f(x) = q as x → p iff any convergent sequence (p_n) s.t. p_n → p w/ p_n ∈ p, p_n ≠ p,
lim f(p_n) = q as n → +inf
F) f: X → Y. f is continuous iff for any p ∈ X', f(p) = lim f(x) as x → p
G) If f: X → Y is continuous, not all open subsets U of X result in f(U) that is also open in Y.
Counterexample: f(x) = x^2
Lecture 13
24. Does Heine-Borel Theorem apply if K is not a subset of R^n?
25. What are examples of subsets that are both open and closed?
A) ex) S ⊂ X. X = R, d(x, y) = |x - y|, S = [0, 1] ⊂ X
Example of open set in S that is not open in X:
(1/2, 1]
26. Why is (1/2, 1] open in S?
27. X = R, S = {1/n: n ∈ N} ∪ {0}. Why is the set {0} not open?
B) Induced topology:
S ⊂ X, E ⊂ S. E is open in S iff ∃ open subset F ⊂ X, s.t. E = S ∩ F
28. How does induced topology graphically look like?
29. Use induced topology to show #27.
C) Inclusion map:
l: S → X, If preserve distance, then l is continuous.
D) Compactness is an intrinsic notion.
E) Compatible topology:
l: X → Y, ∀ U ⊂ X open, ∃ V ⊂ Y open s.t. U = X ∩ V.
For inclusion map w/ compatible topology, if K ⊂ X is compact, K ⊂ Y is compact.
30. If V_a are open, why is V_a ∩ X open in X?
F) f: X → Y is continuous, E ⊂ X is compact. Then, f(E) ⊂ Y is compact.
G) Showing compactness:
Show that there is a finite open cover.
