Real Analysis Answers

3.3 Which of the following statements are True or False. In case(s) where False is the answer 

provide the correct answer. 

3.3.1 No integer is an irrational number 

3.3.2 Every integer is a rational number 

3.3.3Every integer is a whole number

I need help with part 2 of the following question. Please show A separates points and rule out the set C_x0,y0. By Stone's theorem, this would imply uniform density. There's another poster on this site who gave a nonsense answer to this question.

Let X, Y be compact metric spaces. Let A = {(x, y) →summation i=1 to n fi(x)gi(y) | fi ∈ C(X, R), and gi ∈ C(Y, R), 1 ≤ i ≤ n}.

i. Show that A is an algebra.

ii. Show that A is uniformly dense in C(X × Y, R).

the following statements true or false? Give reasons for your answer.

 a) For the function f, defined by f(x) =4x²-4x²- 7x -2,there exists a point 

C ∈ ]-1/2,2[ satisfying f′(c) = 0

 b) For all even integral values of n,

lim (x+1)^-n exists.

x→∞

 c) The function f defined by f(x)= [x − 1], (where [x] is the greatest integer 

function) is integrable on the interval [2,-4].

 d) Every infinite set is an open set. 

 e) All strictly monotonically decreasing sequences are convergent.