(3) Prove that the sequence fn (x(max10,z - n))2 does not converge uniformly on IR, but converges...
(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
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
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].
(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...
For the following statements give a counterexample or demonstrate them: a)If fn (x) is a succession of functions uniformly bounded. Does this suc- cession have a subsucession that converges at least punctually in its domain? b)If {fn (2.)) is a succession of continuous, bounded, defined functions in a compact and that converge punctually in said compact. Is {fn (x)) a succes- sion of functions uniformly bounded? For the following statements give a counterexample or demonstrate them: a)If fn (x) is...
Tamo . Suppose that a sequence of functions fn converges pointwise to a function f on a set E, but there exists a sequence of points In E E such that \fn(2n) – f(2n) > for some strictly positive l. Then fn does not converge uniformly to f on E. (You don't need to prove this here, but it should be clear why this is true.) Now let nar2 fn(L) = 2 +n323 Show that fn converges pointwise on [0,0]...
(5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded and equicontinuous function f E a (5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded and equicontinuous function f E a
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...
(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...
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)?