Suppose A CR and B CR are sets, and that a = sup A and b = sup B exist. (a) Prove that ANB is bounded above by a (and also by b). (b) Suppose ANB+ 0. Prove that ANB has a supremum, m say, and that m < min{a, b}. (c) If An B+ Ø is it necessarily the case that m = proof or give a counterexample. min{a, b}? Either give a

Q: and are measurable fune then f? +2 is measurable func

A: Using the properties of measurable functions,

Q: (11x-3) (4x +39) *Theorem:

A: Explanation of the solution is given below...

Q: For any topological space (X, T) and A C X, the set A is closed. That is, for any set A in a…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Let (X, M, µ) be a measure space such that uX = 1, and let f : X → (0, 1). Show that u) / fdµ 2 1

A:

Q: prove for all x,y in R^n: (1+||x+y||) ≥(1+||x||)/(1+||y||)

A: Sol:- The inequality to be proven is: (1 + ||x + y||) ≥ (1 + ||x||)/(1 + ||y||) Let's start by…

Q: Two nonnegative numbers whose product is 25a² (where a 0) and whose sum is a minimum, are

A:

Q: 1 2. 1+sec¯x

A:

Q: Let x, y be real numbers such that xy=1x and x≠y. Prove that x^2+y^2≥2\sqrt{2}|x−y|.

A: Given: xy=1x,x≠y To find: x2+y2≥22x-y

Q: Show that the identity k n = n 72 n k k-1, olds for all integers 1 <k<n and use it to prove that n…

A:

Q: Vx4 + 39x + 13 – 4 = 0

A:

Q: x2 ax+2 x>4 find the ualue of a for the continuity of f6x) at x=1.

A:

Q: Two nonnegative numbers whose product is 36a? (where a > 0) and whose sum is a minimum, are

A: Now we will solve the problem.

Q: eate a table of integer values for x = -2 through x = 2 for _x)=x²-5

A: Topic:- calculus

Q: how that the quality [x+y]=[x]+[y] is not valid for all real numbers x and y. please explain…

A:

Q: Prove that zero is a root for the equation ax' + bx² +cx+d = 0 if and only if d = 0. What can you…

A:

Q: Zero(s) of multiplicity one: Zero (s) of muit iplicity jwo Żerols) of muitnpncitj inree:

A: multiciplicity is no of times repetation of zeros that is known by putting given equation to zero.

Q: Provide either a proof or a counterexample for  For all positive real numbers x, x^2 - x ≥ 0

A: According to the given information, it is required to prove the given equality or give a counter…

Q: Which quantification proves a nonnegative number minus a negative number is positive? A Vxay(x 0) ^…

A: A ) ∀ x ∃ y(x&lt;y)

Q: Show that the nondegenerate interval [2, 5] is uncountable.

A: Here first we have to show that  is not countable.If possible let,  is countable ,so A can be…

Q: What is the least upper bound of the real interval (2, 6)?

A:

Q: To prove L'Hopital's Rule for the case, you use what Theorem and what statement is a direct result…

A: For ∞∞ case, we can use the l'Hospital rule by taking the derivatives of numerator and denominator…

Q: 4. Prove that the interval (0, 1) does not contain a smallest value x.

A:

Q: Prove algebraically that  A+A'B'=A+B'

A: We have to prove algebraically that  A + A'B' = A + B' We know that, the distributive property  A +…

Q: Use intermediate value theorem to prove that a cubic  x3+ax2+bx+c=0 for at least one real value of…

A: Intermediate value theorem: If f is function which is continuous at every point of the interval…

Q: 9:Find the absolute exlemum Verlues of flx) x²-4x+24 over the interNal Cl,41

A:

Q: How many injective functions have domain [3], codomain [101], and are not surjective?

A: Find the number of injective functions from the domain 3 to the co-domain 101 which is not…