Math 104: Homework 2 (due February 4)

1. Consider each of the following sets:

\begin{matrix}A = (0, ∞), & B = \{\frac{1}{m} + \frac{1}{n} | m, n ∈ \}, & C = \{x^{2} − x − 1 | x ∈ \}, \\ D = [0, 1] ∪ [2, 3], & E = ∪_{n=1}^{∞} [2n, 2n + 1], & F = ∩_{n=1}^{∞} (1 − \frac{1}{n}, 1 + \frac{1}{n}). \end{matrix}

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. Ross exercise 4.11

3. Ross exercise 4.14

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

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

(a) ( \frac{3n}{n+3} )^{2},

(b) \frac{1+2+...+n}{n^{2}} ,

(c) \frac{a^{n}−b^{n}}{a^{n}+b^{n}} for a > b > 0,

(d) n^{2}/2^{n},

(e) \sqrt{n + 1} − \sqrt{n}. Detailed proofs are not required, but you should justify your answers.

6. Ross exercise 7.4

7. Let s_{n} = n!/n^{n} for all n ∈ . Prove that s_{n} → 0 as n → ∞.

8. Let (s_{n})_{n∈} be a sequence such that s_{n} → s as n → ∞. Prove that if p : → is a polynomial function, then p(s_{n}) → p(s). [Hint: Use the limit theorems for addition and multiplication of sequences.]

9. Let s_{1} = t for some t ∈ , and define a sequence according to s_{n+1} = 1 + \frac{s_{n}}{2} for n ∈ . Prove that for all t, s_{n} → 2 as n → ∞.

10. Optional for the enthusiasts. Let s_{1} = t for t ∈ , and define s_{n+1} = \frac{s_{n}}{2}(3 − s_{n}^{2}) for n ∈ . For all values of t, determine and prove whether s_{n} is convergent, divergent but bounded, or divergent and unbounded.