Math 104 homework 2 optional for the enthusiasts let s1 t

Question

Math 104: Homework 2-

1. Consider each of the following sets:

A = (0, ∞),

B = {1/m + 1/n|m, n ∈ N},

C = {x² - x - 1 | x ∈ R},

D = [0, 1] ∪ [2, 3],

E = _n=1∪^∞[2n, 2n + 1],

F = n=1∩∞(1 - 1/n , 1 + 1/n).

For each set, determine it's minimum and maximum if they exist. In addition, determine each set's infimum and supremum, writing your answers in terms of infinity for unbounded sets. Detailed proofs are not required.

2. Let A and B be nonempty bounded subsets of R, and let M = {a · b | a ∈ A, b ∈ B}. Is sup M = (sup A) · (sup B)? Either prove, or provide a counterexample.

3. Find the limits of each of the following sequences, defined for n ∈ N:

(a) (3n/n+3)²,

(b) 1+2+...+n/n²,

(c) aⁿ-bⁿ/aⁿ+bⁿ for a > b > 0,

(d) n²/2n,

(e) √(n + 1) - √n.

Detailed proofs are not required, but you should justify your answers.

4. Let s_n = n!/nⁿ for all n ∈ N. Prove that s_n → 0 as n → ∞.

5. Let (s_n)_n∈N be a sequence such that sn → s as n → ∞. Prove that if p : R → R is a polynomial function, then p(s_n) → p(s). [Hint: Use the limit theorems for addition and multiplication of sequences.]

6. Let s₁ = t for some t ∈ R, and define a sequence according to s_n+1 = 1 + (s_n/2) for n ∈ N. Prove that for all t, s_n → 2 as n → ∞.

7. Optional for the enthusiasts. Let s₁ = t for t ∈ R, and define s_n+1 = s_n/2(3 - s²_n) for n ∈ N. For all values of t, determine and prove whether s_n is convergent, divergent but bounded, or divergent and unbounded.