Lecture Notes

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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:

1) If there are infinite dominant terms, construct subsequence using the dominant terms.

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