Math 104 homework 2 for each set determine its minimum and

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

5. Let (sn)_n∈N be a sequence such that s_n → 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 → ∞.