2. (8 points) Let {fn}n>ı be a sequence of functions that are defined on R by...
1. (a) Let {fn}neN : [0,00) + R be a sequence of function define by: sin(nx) fn(x) 1+ nx (i) Guess the pointwise limit f of fn on (0,00) and justify your claim. [15 Marks] (ii) Show that fn + f uniformly on ſa, 00), Va > 0. [10 Marks) (iii) Show that fn does not converge uniformly to f on (0,00) [10 Marks] (Hint: Show that ||fr|| 21+(1/2) (b) Prove that a continuous function f defined on a closed...
Proposition 7.27. Suppose fn: G + C is continuous, for n > 1, (fn) converges uniformly to f :G+C, and y C G is a piecewise smooth path. Then lim n-00 $. fn = $. . 7.23. Let fn(x) = n2x e-nx. (a) Show that limn400 fn(x) = 0 for all x > 0. (Hint: Treat x = O as a special case; for x > 0 you can use L'Hôspital's rule (Theorem A.11) — but remember that n is...
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...
2. Let a be a positive real number, let r be a real number satisfying r >1, let N be an integer greater than one, and let tR -R be the integrable simple function defined such that tr,N(r) = 0 whenver x < a or z > ar*, tr,N(a) = a-2 and tr,N(z) = (ar)-2 whenever arj-ıく < ar] for some integer j satisfying 1 < j < N. Determine the value of JR trN(x) dz.
Many thanks!! (a) Let fn(x) max(1 - |x -n|,0) for each n 2 1. Show that {fn} is a bounded sequence in LP (R) for all p E [1, 00]. Show that fn >0 pointwise everywhere in R, i.e. fn(x) -> 0 for all x E R. Show that fn does not converge to 0 in LP (R) (b) Fix p E 1, o0). Let fn E LP(0, 1) be defined by fn(x) n1/? on [0,1/n), and fn(x)0 otherwise. Show...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
(Exercise 9.2) Let f,, : R → R, fn(x)-n and f : R → R, f(x) fn does not converge uniformly to f (i.e. fn /t f uniformly) 0. Prove that fn → f pointwise but (Exercise 9.2) Let f,, : R → R, fn(x)-n and f : R → R, f(x) fn does not converge uniformly to f (i.e. fn /t f uniformly) 0. Prove that fn → f pointwise but
(10) Suppose that fn : R -> R is a sequence of functions such that for every xo E R there exist a neighborhood N(ro) of xo on which fn converges uniformly. (a) Prove that fn converges uniformly on every compact subset of R (10) Suppose that fn : R -> R is a sequence of functions such that for every xo E R there exist a neighborhood N(ro) of xo on which fn converges uniformly. (a) Prove that fn...
(3) Prove that the sequence fn (x(max10,z - n))2 does not converge uniformly on IR, but converges uniformly on compact subsets of R (3) Prove that the sequence fn (x(max10,z - n))2 does not converge uniformly on IR, but converges uniformly on compact subsets of R
5b. (5 pts) Let fn : [0, 1] - R be given by I fn (2) = 1 n²s if 0 2TO 2n-nar if < 0 if < < < 1 Find limno Sofr (x) dx and Slimnfr () dx and use it to show that {fn} does not converge uniformly. Justify your answer.