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Someone can help? For each n = 1,2,3,..., define fn (2):= (1 – 22n), for every...
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x E A, we have lÍs(x)-f(x)|く for all n > N. Then f is uniformly continuous. Note: We could say that the "sequence of functions" f "converges to the function" f. These are not defined terms for...
Problem 2 1. Let fn(ar) n As the metric take p(x, y) = |x - y. Does lim, fn(x) exist for all E R? If it exists, is the convergence uniform. Justify 2. Consider fn(x) = x2m, x E [0, 1]. Is it true that lim (lim fn(= lim( lim fn(x)) noo x-1 Justify.
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove that for any A E A f du lim fn du A 4 (You must show that the integrals exist.)
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove...
4. Suppose (fr)nen is a sequence of functions on [0, 1] such that each fn is differentiable on (0,1) and f(x) < 1 for all x € (0,1) and n e N. (a) If (fn (0))nen converges to a number A, prove that lim sup|fn(x) = 1+|A| for all x € [0, 1]. n-too : (b) Suppose that (fr) converges uniformly on [0, 1] to a function F : [0, 1] + R. Is F necessarily differentiable on (0,1)? If...
part (c)
7.23. Let y(x) = n²x e-nx. (a) Show that lim, - fn(x)=0 for all x > 0. (Hint: Treat x = 0 as for x > 0 you can use L'Hospital's rule (Theorem A.11) - but remember that n is the variable, not x.) (b) Find lim - So fn(x)dx. (Hint: The answer is not 0.) (c) Why doesn't your answer to part (b) violate Proposition 7.27 Proposition 7.27. Suppose f. : G C is continuous, for n...
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1) Let F be the function from R x (-1,1) to R3 given by F(u,0)= ( (2- sin u, vsin (2+v cos vcos COS u Let (u, ) and (u2, 2) belong to the domain R x (-1, 1) of F. Prove that F(u1, U1) (u1(4k 2),-v1) for some relative integer k. Hint: In terms of the spacial coordinates a, y,z compare the quantities 2 +y2 F(u2, 2) if...
CR, we typically think of t if : >0.. 1-1 if : <o'' this is the natural way we might define the 'magnitude of a real number, but it is not the only way. a.) Prove that for ry ER, we have xy = 13. lyl. b.) Construct a new function : R-R UO) such that for r, y € R, we have: 1.) ||2||=0- I = 0 and ii.) ||3+ yll |||| + llyll but iii.) xyll ||||llyll. 36....
I just need 3d answered please!
(3) The Hypergeometric Function If a, b, c R with c f {0, -1,-2,...^ we define the Gauss hypergeometric function as n!c(c 1)... (c+n-1) Note that this solves the DE (a) Verify that log(1x) rF(1,1,2, -) (b) Verify formally (without justifying the limits) that e-lim F (a, b, a, (c) Show that Pla, b, c, x) = abF(a + 1,D+ 1, c + 1, x) (d) Show that F(n, -n, s a polynomial, and...
Question 2. a) The zero transformation. We define the zero transformation, To: FN → Fm by To(x) = 0 VxEFN. (i) What is R(To)? (ii) Is To onto? (iii) What is N(To)? (iv) Is To one-to-one? (v) What is (To]s? b) The identity transformation. We define the identity transformation, Tj: Fn + En by Ty(x) = x V xEFN. (i) What is R(Ti)? (ii) Is T, onto? (iii) What is N(T)? (iv) Is T one-to-one? (v) What is Ti]s? Question...