Real Analysis Answers

If u be a borel measure on a metric space X and suppose there exists a sequence Gn of open sets, X=UnGn and for all n , u(Gn) < Infinity then u is regular.

Prove that the set of complex numbers is field.

borel uniqueness theorem

a) Let D be a nonempty subset of R, bounded above. Is M = sup(D) a limit point of D? Explain.

b) Let the function g : R → R be bounded. Define f(x) = 1 + 4x + x^2(g(x)). Prove that

f(0) = 1 and f'(0) = 4. (You may not assume g is differentiable.)

c) Suppose that the function f : R → R is differentiable and that there is a bounded sequence {xn} with xn doesnt= xm if n doesnt= m, such that f(xn) = 0 for every index n. Show that there is a point x0 at which f(x0) = 0 and f'(x0) = 0. (Hint: Use Bolzano-Weierstrass Theorem.)

Suppose that (𝑥∝)∝𝜖𝐽 ⟶ 𝑥 in X and (𝑦∝)∝𝜖𝐽 ⟶ 𝑦 in Y. Show that

(𝑥𝛼 × 𝑦𝛼) → 𝑥 × 𝑦 𝑖𝑛 𝑋 × 𝑌.

Let I1, I2 ⊂ R be two bounded open intervals in R. Prove tht

I1 × I2 = { (a, b) : a ∈ I1, b ∈ I2 } ⊂ R2

is an open subset of R2