Let (fn) and (9n) be two sequences of functions [a, b] + R, each of which converges uniformly, lim fn = f, lim 9n = g. Suppose that f and g are bounded. Show that then, (Anon) also con- verges uniformly to fg. Please write your solution to this problem out clearly in LaTeX
(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
Suppose that fn(x) converges to f(x) uniformly, that the functions fn(x) are all differentiable, and that the function f(x) is also differentiable. (All of these conditions are assumed to be true on a bounded, closed interval [a, b].) Prove or disprove: lim as n goes to infinity fn'(x) = f'(x)
Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if and only if fn(xn) → f(x) whenever xn → x. Suppose that functions fn : [0, 1] → R, for n = 1,2. . . ., are continuous and f : [0, 1] → R is also continuous. Show that fn → f uniformly if...
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.
(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
10 Let fn be a sequence of functions that converges uniformly to f on a set E and satisfies IfGİ M for all 1,2 and all r e E. Suppose g is a continuous function on [-MI, M]. Show that g(Um(x)) uniformly to g(f(r)) on E 10 Let fn be a sequence of functions that converges uniformly to f on a set E and satisfies IfGİ M for all 1,2 and all r e E. Suppose g is a continuous...
(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 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...
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...