clear and clean answer,pls 2. Suppose S and T are nonempty sets of real numbers with the following property: (P) There...
2. Suppose S and T are nonempty sets of real numbers with the following property: (P) There exists y in T such that for all x in S, y>x. Prove the following (of course, use the standard approach for proving universally and existentially quantified statements): (Q) For all x in S, there exists y in T such that y>x.
1. Suppose that a function f, defined for all real numbers, satisfies the property that, for all x and y, f(x + y) = f(x) + f(y). (*) (a) (2 points) Name a function that satisfies property (*). Name another that doesn't. Justify your answers. (b) (3 points) Prove that any function that satisfies property (*) also satisfies f(3x) = 3f(x). (c) (5 points) Prove that any function that satisfies property (*) also satisfies f(x - y) = f(x) –...
1. Suppose that a function f, defined for all real numbers, satisfies the property that, for all x and y, f(x + y) = f(x) + f(y). (*) (a) (2 points) Name a function that satisfies property (*). Name another that doesn't. Justify your answers. (b) (3 points) Prove that any function that satisfies property (*) also satisfies f(3x) = 3f(x). (c) (5 points) Prove that any function that satisfies property (*) also satisfies f(x - y) = f(x) –...
(a) Find (22,8,P) and nonempty sets (An)>1 C 8 such that P (liminf An) <liminf (P(Ar)) < limsup(P (A)) <P (limsup An). (b) Given A, B,(A.)21,(B.).>1 C , either prove the following statements or find counterexamples to them. i. limsupA, U limsupB = limsupA, UB.. ii. liminf Anu liminfB = liminfAUB iii. limsupA, nlimsupB. = limsupA, B. iv. liminfa, nliminfB = limin A, B, (e) Prove that probability spaces have the property of continuity from above. (We proved continuity from...
5. Describe the following sets of real numbers and find the supremum and infimum of these sets: (a) {x}\x2 – 2 <4€R} (b) {x|x+ 2 +13 – x4<4} (©) {x|x<for all neN} 6. For any two elements x and y of an ordered field, prove that _x+ y + x- x + y - x - y (a) max{x,y}=- (b) min{x,y}=-
1. Suppose that P is the uniform distribution on [0,1). Partition the interval [0,1) into equivalence class such that x ~ y (x is equivalent to y) if x-y є Q, the set of rational numbers 2. Given 1, by the Axiom of Choice, there exists a nonempty set B C [0,1) such that IB contains exactly one member of each equivalence class. Prove each of the following (a) Suppose that q E Qn [0, 1). Show that B (b)...
Suppose that the functions s and t are defined for all real numbers x as follows. s(x)=x-6 t(x)= 3x2 Write the expressions for (s – t)(x) and (s·t)(x) and evaluate (s+t)(-3). (s – t)(x) = _______ (s·t)(x) = _______ (s + t)(-3) = _______
Suppose that the functions s and t are defined for all real numbers x as follows. (r)-5r t(r) 3x-2 Write the expressions for (+s)(x) and (-)) and evaluate (t s)X-2). +3)-0 ts(x) s)(x) = (rs)(-2)D
Real analysis
10 11 12 13 please
(r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....
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2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...