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-**Lecture 4/6**
+**Questions + Notes**
-Differentiability (f: [a, b] -> R)
+**Lecture 1**
-) f is differentiable at p ∈ [a, b] if:
+. Why is |sin(nx)| <= n|sin(nx)| ∀ n ∈ N, ∀ x ∈ R?
-   lim ((f(x) - f(p))/(x - p)) as x -> p exists.
+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:
-) f'(p) = lim ((f(x) - f(p))/(x - p)) as x -> p
-If f is differentiable at p ∈ [a, b]:
+d | c_n, c | c_0 (i.e. factors of constant/factors of leading coefficient is a solution to the equation)
-) f is **continuous** at p.
+B) Completeness axiom
-) ∃ u(x) s.t. f(x) = f(p) + (x - p)f'(p) + (x - p)u(x)
+If S ⊂ R is bounded from above, sup(S) exists in R
-   *lim u(x) = 0 as x -> p
+If S ⊂ R is bounded from below, inf(S) exists in R
-Let g: [a, b] -> R also be differentiable at p ∈ [a, b]:
+**Lecture 2**
-) (f + g)'(p) = f'(p) + g'(p)
+. For -S = {-x | x ∈ S}, why is -S bounded above, and why is inf(S) = -sup(-S)?
-) (f⋅g)'(p) = f'(p)g(p) + f(p)g'(p)
+. How doe we show that lim a_n = 0 as n -> +inf given a_n = sin(n)/n **using definition of limit**?
-) (f/g)'(p) = (f'g - fg')/(g^2) if g(p) ≠ 0
+A) If max(S) = sup(S), inf(S) = min(S), S is connected:
-Let f, g: R -> R, f(a) = b, and f is differentiable at a, g is differentiable at b:
+S is a closed (bounded) interval
-) h(x) = g(f(x)) is differentiable at a; h'(a) = g'(f(a))·f'(a)
+B) Checking that sup(S) = M:
-Let f, g: [a, b] -> R:
+Step 1: Check that M is an upper bound of S
-) p ∈ [a, b] is a local maximum (**local minimum**) of f if there is a δ > 0 s.t.
+Step 2: Check that ∀ α < M, α is not an upper bound of S
-   ∀ x ∈ [a, b] ∩ Ball with center at p,
+C) Archimedian Property:
-   we have f(x) ≤ f(p) (**f(x) ≥ f(p)**)
-) If p is a local maximum (or local minimum) of f, p ∈ (a, b), f'(p) exists:
-    f'(p) = 0
+If a, b > 0, then ∃ n ∈ N s.t. na > b
-) If f, g are differentiable on (a, b), ∃ c ∈ (a, b) s.t.
-    [f(b) - f(a)]g'(c) = [g(b) - g(a)]f'(c)
+**Lecture 3**
-  * If g(x) = x,
-    f(b) - f(a) = (b-a)f'(c)
-Let f: [a, b] -> R, f is continuous, f'(x) exists ∀ x ∈ (a, b), and
-f(a) = f(b):
+. 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)
-) ∃ c ∈ (a, b) s.t. f'(c) = 0
+. 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**
+. Why is lim x^(1/x) = lim e^((log x)/x)? (as x -> +inf)
+. 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**
+. 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**
+. How do you construct a polynomial equation with sqrt(2 + sqrt(2)) as its root?
+. Why are we able to find the limit of a recursive sequence using the "zig zag trajectory"? (Refer to (B) below)
+. 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**
+. 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.
+. 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**
+. 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)
+. 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**
+. Why is S = R \ {0} non-complete?
+. Why is lim(s_n) = (s_n)_n if (s_n) is Cauchy?
+. 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**
+. Prove that the closure of E is the union of E and E'.
+. 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?
+. 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**
+. 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?
+. 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**
+. Does Heine-Borel Theorem apply if K is not a subset of R^n?
+. 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]
+. Why is (1/2, 1] open in S?
+. 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
+. How does induced topology graphically look like?
+. 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.
+. 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))
+. 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**
+. Show that sinx is uniformly continuous.
+. 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) = ∅.
+. 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.
+. 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.
+. 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**
+. 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.
+. Describe the difference among Pointwise convergence, d2 convergence, and d∞ convergence.
+. 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.
+. 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)
+. Why is K not compact in the above example?
+B) Open and closed are relative notion, but compactness is an absolute notion.
+. 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.
+. Give examples of (C) in which continuity does not preserve openedness that is not f(x) = x^2 and the domain is (-1, 1).
+. Show f(x) = sin(1/x) is not uniformly continuous.
+. How does f_n -> f uniformly translates to lim sup|f_n(x) - f(x)| = 0 as n -> +inf and for x ∈ X?
+. 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**
+. "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.
+. 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.
+. 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
+.
+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.

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