**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 ((a_n - α)β + α(b_n - β) + (a_n - α)(b_n - β))"?

**Lecture 4**

6. Why is lim x^(1/x) = lim e^((log x)/x)? (as x -> +inf)

7. For S_N = sup {a_n | n >= N}, why is S_N >= S_M for N < M?

A) Prove lim a_n = 1 given a_n = n^(1/n):

Show that lim S_n = 0 when S_n = n^(1/n) - 1

B) All bounded monotone sequences are convergent.

C) If A ⊃ B, sup A >= sup B, inf A <= inf B

D) (a_n) is a Cauchy sequence if ∀ ε > 0, ∃ N > 0 s.t. ∀ n, m > N, |a_n - a_m| < ε

E) (a_n) is a Cauchy sequence iff (a_n) converges.

F) (a_n) converges iff limsup(a_n) = liminf(a_n)

**Lecture 5**

8. In the proof of the theorem "(a_n) is Cauchy iff (a_n) converges," how do we know that liminf(a_n) <= limsup(a_n)?

A) limsup is not a sup of any set (it is limit of sups).

B) To prove a = b, we can show that |a - b| < ε ∀ ε > 0

**Lecture 6**

9. How do you construct a polynomial equation with sqrt(2 + sqrt(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.

H) Sequential compact:

∀ (y_n) in f(E), ∃ (y_n_k)_k s.t. lim y_n_k = y ∈ f(E) as k -> +inf

I) If f: X -> R is continuous and E ⊂ X is compact, there exist p, q ∈ E s.t. f(p) = sup(f(E)), f(q) = inf(f(E))

31. How do we know K = (0, 1] is closed in (0, +inf)?

J) Heine-Borel Theorem applies when X = R^n

H) Pre-image of compact set may not be compact.

Ex: f(x) = 1/x, Image = [0, 1]

**Lecture 14**

32. Show that sinx is uniformly continuous.

33. Why is [0, 2π) not compact? (Referring to an example showing that if X is not compact, we cannot make a conclusion that if f: X -> Y is continuous and f is a bijection, the inverse is also continuous)

A) Uniformly continuous (f: X -> Y):

∀ ε > 0, ∃ δ > 0 s.t. for all p, q ∈ X, d_X(p, q) < δ -> d_Y(f(p), f(q)) < ε

- One delta value works for all p ∈ X (unlike regular continuity)

B) sinx is uniformly continuous, but x^2 is not

C) If f: X -> Y is continuous and X is compact, f is uniformly continuous.

D) If f: X -> Y is continuous, S ⊂ X, then f|_S: S -> Y is continuous.

E) If f: X -> Y is continuous, X is compact, and f is a bijection, then f^-1: Y -> X is continuous.

F) If f: X -> Y is uniformly continuous and S ⊂ X with the induced metric, then f|_S: S -> Y is also uniformly continuous.

G) **Connected** space:

The only subset of X that is both open and closed are X and ∅.

H) If f: X -> Y is continuous, X is connected, then f(X) is connected.

I) If f: X -> Y is continuous, E ⊂ X is connected, then f(E) is connected.

**Lecture 15**

A) Connected subset cannot be written as A ∪ B, where (closure of A) ∩ B = ∅ and A ∩ (closure of B) = ∅.

34. If the subset can be written as a union of A and B (the situation described above), how do we know that A, B are both open and closed in the subset?

A: (closure of A) ∩ S = (closure of A) ∩ (A ∪ B) = ((closure of A) ∩ A) ∪ ((closure of A) ∩ B) = A ∪ ∅ = A. -> A and B are closed.

A and B are complements. Thus, A and B are open.

B) E is connected iff for all x, y ∈ E, and x < y, [x, y] ⊂ E.

35. Why does being a x ∈ X being a limit point of E ⊂ X imply that all (B_ε(x) \ {x}) ∩ E ≠ ∅?

C) If f: (a, b) -> R is a monotone increasing function, f has at most countably many discontinuities.

36. If f: [0, 1] -> R is continuous and f([0, 1]) ⊂ [0, 1], there exists x ∈ [0, 1] s.t. f(x) = x. Prove this statement.

**Lecture 16**

37. Give an example of a function that is pointwise convergent, but not uniformly convergent.

A) Pointwise convergence (f_n: X -> Y):

For all x ∈ X, lim f_n(x) = f(x) as n -> +inf

B) Pointwise limit of a function does not preserve integral.

38. Describe the difference among Pointwise convergence, d2 convergence, and d∞ convergence.

39. How is d∞-metric sense convergence related to uniform convergence?

A: f_n -> f uniformly iff lim d∞(f_n, f) = 0 as n -> +inf

**Lecture 17**

A) Uniformly continuous:

for all ε > 0, there exists N > 0 s.t. for all n > N and for all x ∈ X, we have |f_n(x) - f(x)| < ε.

- N only depends on ε, not on x.

B) Uniformly Cauchy iff Uniformly convergent.

C) f_n: X -> R, 0 <= M_n ∈ R s.t. M_n >= sup|f_n(x)| where x ∈ X.

If an infinite series of M_n from n = 1 to n = +inf is less than +inf, then the infinite series of f_n from n = 1 to n = +inf converges uniformly.

i.e. If the absolute value of f_n(x) is bounded (let's say the bound is M_n), and the infinite sum of M_n converges, then the infinite sum of f_n converges uniformly.

D) Uniform convergence preserves continuity.

E) To show that a function f is continuous, it suffices to show that for all x ∈ X', we have lim f(t) = f(x) as t -> x.

40. Why can we state the above (E)?

F) If K is a compact metric space, f_n: K -> R, f_n is continuous, f_n -> f, f is continuous, and f_n(x) >= f_(n+1)(x),

f_n -> f uniformly.

**Lecture 18**

A) K ⊂ X is compact iff K is closed and bounded **if X = R^n**

Counterexample when X ≠ R^n:

X = (0, 1), K = (0, 1)

41. Why is K not compact in the above example?

B) Open and closed are relative notion, but compactness is an absolute notion.

42. If K ⊂ X is compact and E ⊂ X is closed, why are we able to conclude that K ∩ E is compact?

C) Continuity preserves compactness and connectedness, but not necessarily openedness and closedness.

43. Give examples of (C) in which continuity does not preserve openedness that is not f(x) = x^2 and the domain is (-1, 1).

44. Show f(x) = sin(1/x) is not uniformly continuous.

45. How does f_n -> f uniformly translates to lim sup|f_n(x) - f(x)| = 0 as n -> +inf and for x ∈ X?

46. Show that f_n(x) = x/n converges to 0 pointwise, but not uniformly.

D) E ⊂ X is dense iff (closure of E) = X

**Lecture 19**

A) If f is differentiable at p ∈ [a, b], then f is continuous at p.

B) If f(x) is differentiable at p, then there exists u(x) s.t. f(x) = f(p) + (x-p)f'(p) + (x-p)u(x) where lim u(x) = 0 as x -> p

u(x) = (f(x) - f(p))/(x - p) - f'(p) when x ≠ p, u(x) = 0 when x = p

C) p can be a local maximum, but f'(p) is non existent (cusp).

D) Local maximum and minimum can occur at endpoints.

E) f: [a, b] -> R is a continuous function. Assume f'(x) exists for all x ∈ (a, b). If f(a) = f(b), then there exists c ∈ (a, b), s.t. f'(c) = 0. (Rolle's Theorem)

F) If f, g: [a, b] -> R are continuous and differentiable on (a, b), then there exist c ∈ (a, b) s.t. [f(b) - f(a)]g'(c) = [g(b) - g(a)]f'(c).

- Special case:

f(b) - f(a) = (b-a)f'(c), c ∈ (a, b)

**Lecture 20**

47. "If c is a local maximum of a function, and if derivative exists at c, and c is an interior point, derivative vanishes." How does this statement hold true?

A) f: R -> R, f is continuous, f'(x) exists for all x ∈ R. Assume there exists M > 0 s.t. |f'(x)| <= M for all x. Then, f is uniformly continuous.

B) f: [a, b] -> R is a differentiable function. f'(a) < f'(b). Then, for any c ∈ R w/ f'(a) < c < f'(b), there exists a d ∈ (a, b) s.t. f'(d) = c

C) lim (f(x)/g(x)) = C as x -> a if:

f, g: (a, b) -> R are differentiable, g(x), g'(x) ≠ 0 over (a, b) AND

lim (f'(x)/g'(x)) = C AND

i) lim f(x) = 0 as x -> a, lim g(x) = 0 as x -> a OR

ii) lim g(x) = +inf as x -> a

**Lecture 21**

A) Smooth functions:

Derivatives exist to all order.

B) Taylor Theorem:

f: [a, b] -> R is a function s.t. f^(n-1) (x) exists and is continuous on [a, b]. f^(n) (x) exists on (a, b).

Then, for any α, β ∈ [a, b], we have:

f(β) = f(α) + f'(α)(β-α) + f''(α)/2!(β-α)^2 + ... + f^(n-1)(α)/(n-1)!(β-α)^(n-1) + R_n(α, β)

R_n(α, β) = 0 if α = β, R_n(α, β) = f^n(r)/n!(β-α)^n if α ≠ β for some r ∈ (α, β)

**Lecture 22**

A) Power series:

Series of the form ∑ C_n(x-x_0)^n from n = 0 to n = +inf

B) Radius of Convergence R:

R = sup {r >= 0, s.t. if |x-x_0| <= r, the series converges}

C) If R = 1/a, where a = limsup|C_n|^(1/n) as n -> +inf:

If |x-x_0| < R, the series converges.

If |x-x_0| > R, the series diverges.

48. Prove the diverging case of (C).

D) Real Analytic function f:

f: (a, b) -> R is smooth, for all x_0 ∈ (a, b), f(x) = ∑ C_n(x-x_0)^n for x in some neighborhood of x_0.

49. Regarding the definition of real analytic function, what does it mean for "f(x) = ∑ C_n(x-x_0)^n for x in some neighborhood of x_0"?

E) Length of a function:

∫sqrt(1 + (f'(x))^2)dx

F) U(P, f) = ∑ M_i*Δx_i from i = 1 to i = n

L(P, f) = ∑ m_i*Δx_i from i = 1 to i = n

M_i = sup{f(x)|x ∈ [x_(i-1), x_i]}

m_i = inf{f(x)|x ∈ [x_(i-1), x_i]}

Δx_i = x_i - x_(i-1)

G) U(f) = inf U(P, f)

L(f) = sup L(P, f)

H) f is integrable if U(f) = L(f)

I) Generalization of Riemann-Stieltjes integrable:

Let α: [a, b] -> R be a monotone increasing function, define partition P = {a = x_0 <= x_1 <= ... <= x_n = b}, define Δα_i = α(x_i) - α(x_(i-1))

The remaining definitions are similar as in parts (F) and (G), except Δx_i -> Δα_i, and U(P, α) = L(P, α) implies that f is Riemann-Stieltjes integrable w.r.t. α.

**Lecture 23**

A) If a partition Q is a refinement of partition P on [a, b], then L_P <= L_Q <= U_Q <= U_P.

B) L(f, α) <= U(f, α)

C) f is integrable w.r.t. α iff for all ε > 0, there exists P partition s.t. U_P - L_P < ε

50. 

General Note:

A) Contradiction is very useful in proofs.

When using contradiction, you can select an arbitrary element in a set and prove if it actually belongs to a set.