(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...
Analysis: Give two examples where if fn does not converge to f uniformly on E, but does converge to f pointwise on E, then the following two theorems do not hold. Write clearly and explain and proof your claims. 711 Theorem Suppose fn→f uniformly on a set E in a metric space. Let x be a limit point of E, and suppose that (15) Then (A,) converges, and (16) lim f()im A In other words, the conclusion is that lim...
4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {fn} converges pointwise. fn. Does {6 fn} converge to (b) For each n EN compute (c) Can the convergence of {fn} to f be uniform? 4. For each n EN let fn: [0,1]R be given by if xE(0, otherwise fn(x) = (a) Find the function f : [0, 1] R to which {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
3. For each n E N let fn : (1, 0) -+ R be given by f/(x) = Find the function f : (1, 0) - R to which {fn} converges pointwise. Prove that the convergence is not uniform 3. For each n E N let fn : (1, 0) -+ R be given by f/(x) = Find the function f : (1, 0) - R to which {fn} converges pointwise. Prove that the convergence is not uniform
Let fn be continuous on [a, b] for each n and let fn converge uniformly on (a,b) n Q. Prove that fn converges uniformly on [a, b].
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...
Let f, (x) := lxl1+1/n, Π ε N, and f(x) 비파 Show Exercise 13: a) fn-f uniformly on all bounded intervals (a, b) C R. b) fn -f is not uniformly on all of R. Let f, (x) := lxl1+1/n, Π ε N, and f(x) 비파 Show Exercise 13: a) fn-f uniformly on all bounded intervals (a, b) C R. b) fn -f is not uniformly on all of R.
(b) Let a >0. Does (f.) converge uniformly on [-a, al? (c) Does (f) converge uniformly on R? Q4 You are given the series n2 +r2 (a) Prove that the series converges uniformly on [-a, al for each a > 0. (b) Prove that the sum F(r) is well defined and continuous on R. (c) Prove that the series does not converge uniformly on R. Q5 You are given the series I n2r2 (b) Let a >0. Does (f.) converge...
plz help me analysis question! Thanks in advance 5. For each n є N let fn : R R be given by f,(x)-imrz. Prove that the sequence {f. of functions converges pointwise to the function f R- R given by 1+nr if x#0 f(x)-0 5. For each n є N let fn : R R be given by f,(x)-imrz. Prove that the sequence {f. of functions converges pointwise to the function f R- R given by 1+nr if x#0 f(x)-0
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...