Question 5 For any r E R, prove that there exists q E Z such that q – 1 < x < q.

Q: Prove that for all a € R, 0 < r² + 2 VI

A: Prove the following

Q: Let > 0 and 6 > 0 be real numbers that satisfy e' (1+a+²) = eª (1+b+ 2 ) eb Prove or disprove that a…

A: Its given that a&gt;0 and b&gt;0. We have to prove or disprove that eb1+a+a22=ea1+b+b22 is satisfied…

Q: Use Theorem 2 to show that 3, if x <-10 2, if – 10 sxso F:R → R,F(x) = -2, if 0< x< 10 -3, if x2 10…

A: so we have to show that given function is discontinuous at 2 and -2 (a)   Discontinuous  at -2…

Q: Let n > 3 where x, y , z € R", |x – y| = d > 0, and r is some positive real number. Prove that if 2r…

A: It is given that x and y are d distance apart and r&gt;d/2 where x, y, z belongs to Rn So locus z…

Q: ) If ƒ = C([0, ∞)) n D ((0, ∞)), f(0) = 0 and ƒ' is increasing, then f(x) X f(y) - Y

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Q: For any x ER, prove that there exists q E Z such that q-1<r <q.

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Q: Let x, y ∈ R such that x 0.001. Prove that |x · y| = |x| · |y|

A: Given: x, y∈ℝ, x&lt;-0.001 and y&gt;0.001   To prove: x·y=x·y

Q: Let x, y EZ. Prove that 3|xy if and only if 3|x x or 3 y.

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Q: Prove that if 0 < a < y, then ỹ – VI < Vy– a.

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Q: (c) Let n be a positive integer and assume that there are integers a and b ab. Show that a < yn or b…

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Q: 4. Let f be defined for all real x, and suppose that \fx) – f(v)| < (x – y)? for all x, y e R. Prove…

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Q: 6. Prove that |xy| = |x|ly| for all real numbers x, y.

A: The given problem is to prove the following statement. Given statement is |xy|=|x||y| for all real…

Q: Show that if x,y,z,w are real numbers such that x < y and z ≤w, then x + z < y + w.

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Q: 2. Prove that, for all x ≤ R, | sinx| ≤ |x|. E

A: To given solution-Explanation:Hence proved.

Q: 5. Prove that if ƒ : [-1, 1] → R is measurable, then |f| is also measurable.

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Q: Let n E N coprime with p. Show that there exists x, y E Z coprime and such that p | x² + ny² is and…

A: To prove the given claim we have to show that :- x,y ∈ Z ; gcd(x,y)=1 ; p | x2 + ny2   implies…

Q: If a € R such that 0 0, then show that a = 0.

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Q: Hallar a de modo que h sea continua para todo ax²-x, x≤-1 ax - 5, x>-1 h(x) =m Envie sus comput X…

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Q: Let x ∈ R. Prove that if x 0.

A: Given :

Q: Let x € Z. Prove that if 2 | (x²-5) then 4 | (x²-5).

A: Given that .The objective, if then prove that .

Q: Prove that X and Y are independent if and only if R = 0

A: To prove that if x and y are independent then correlation between them R =0

Q: Prove that e" > x+ 1, for all positive real numbers a.

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Q: Let x ∈ R. Prove that if x3 − x > 0 then x > −1.

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Q: By algebraic properties of R, prove that for every x, y € R with y # 0 (-x) y

A: We have to prove that by algebraic property that ∀x,y∈ℝ with y≠0 -xy=-xy

Q: Let x, y € Z. Prove that if 2 | (x² - y2) then 4 | (x² - y²)

A: We know that the sum of two odd numbers is an even number.Also, the difference between two odd…