Reference no: EM132619238

Real Analysis - Assignment

Question 1. Prove that √12 is an irrational number.

Question 2. Prove that n < 2ⁿ for all n ∈ N.

Question 3. Suppose S be a nonempty set of real numbers which is bounded below and -S be the set of all numbers -s, where s ∈ S. Show that

inf S = - sup(-S).

Question 4. A number x is called an algebraic number if there exists an n ∈ N a₀, a₁, ..., a_n ∈ Z such that

a₀ + a₁x + ... + a_nxⁿ = 0.

Show that every rational is an algebraic number. Are √2, 7^1/3 algebraic numbers? Justify your answers.

Question 5. Prove the following result: Suppose a rational p/q where (p, q) = 1 solves the equation

a₀ + a₁x + ... + a_nxⁿ a₀ ≠ 0, a_n ≠ 0

where n ≥ 1 and a_k ∈ Z for 0 ≤ k ≤ n. Then
• p divides a₀, i.e., a₀ = p.k for some integer k ∈ Z.
• q divides a_n, i.e., a₀ = p.m for some integer m ∈ Z.

Question 6. Apply above theorem to show that there is no rational root for the following equation x² - 2 = 0.