Many thanks!! (a) Let fn(x) max(1 - |x -n|,0) for each n 2 1. Show that...
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}...
(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
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...
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
2. (8 points) Let {fn}n>ı be a sequence of functions that are defined on R by fn(x):= e-nx. Does {{n}n>1 converge uniformly on [0, 1]? Does it converge uniformly on (a, 1) with 0 <a<1? Does it converge uniformly on (0, 1)?
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
= 5a. (10 pts) Let fr : [0, 1] → R, fn(x) ce-nzº, for m = = 1, 2, 3, .... Check if the sequence (fn) is uniformly convergent. In the case (fr) is uniformly convergent find its limit. Justify your answer. Hint: First show that the pointwise limit of (fr) is f = 0, i.e., f (x) = 0, for all x € [0, 1]. Then show that 1 \Sn (r) – 5 (w) SS, (cm) - Vžne 1...
Please answer it step by step and Question 2. uniformly converge is defined by *f=0* clear handwritten, please, also, beware that for the x you have 2 conditions , such as x>n and 0<=x<=n 1- For all n > 1 define fn: [0, 1] → R as follows: (i if n!x is an integer 10 otherwise Prove that fn + f pointwise where f:[0,1] → R is defined by ſo if x is irrational f(x) = 3 11 if x...
b) (10 pts) Let D(0, oo)) be the vector space of all bounded continuous functions from [0, oo) such that R If(x) dz 00. Give an example of a sequence {fn} of functions in D(0,00)) which (i) converges pointwise for E [0, oo) to the constant function f(z)0 (ii) does not converge to 0, neither with respect to the norm, nor the Hint: it may be helpful to contemplate the phrase "mass escaping to infinity". norm. b) (10 pts) Let...
Let fn (x) = 1 + (nx)? {n} are differentiable functions. (a) Show that {fn} converges uniformly to 0. (b) Show that .., XER, NEN. converges pointwise to a function discontinuous at the origin.