Real Analysis: For each n є N and each x є [0,1j, define fn(x)-nxe-nz2 . Find...
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
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...
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...
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...
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...
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. 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
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...
Let (X, dx), (Y, dy) be metric spaces and fn be a sequence of functions fn: XY Prove that if {fn} converges uniformly on X then for any a є x lim lim fn()- lim lim /) xa n-00 Let (X, dx), (Y, dy) be metric spaces and fn be a sequence of functions fn: XY Prove that if {fn} converges uniformly on X then for any a є x lim lim fn()- lim lim /) xa n-00
= 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...