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(10) Prove that if (fn) is a sequence of uniformly continuous functions on the interval (a,...
Let (X, dx), (Y, dy) be metric spaces and fn be a sequence of functions fn: XY Prove that if {fn} converges uniformly o...
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
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...
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 (, A, ) be a measure space. Let fn : 2 -» R* be a sequence of measurable functions. Let g,h : 2 -» R* be a pair of...
Let (, A, ) be a measure space. Let fn : 2 -» R* be a sequence of measurable functions. Let g,h : 2 -» R* be a pair of measurable functions such that both are integrable on that a set A E A and g(r) fn(x)h(x), for all E A and nE N. Prove / fn du lim sup A lim inf fn dulim inf lim sup fn du A fn du no0 no0 A noo n+o0 (You may...
Suppose that fn(x) converges to f(x) uniformly, that the functions fn(x) are all differentiable, and that...
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)
(6) Let (, A,i) be a measure space. Let fn : 0 -» R* be a sequence of measurable functions. Let g, h : O -> R* be a...
(6) Let (, A,i) be a measure space. Let fn : 0 -» R* be a sequence of measurable functions. Let g, h : O -> R* be a pair of measurable functions such that both are integrable on a set A E A and g(x) < fn(x)<h(x), for all E A and ne N. Prove that / lim sup fn du fn dulim sup fn du lim inf fn du lim inf n o0 A n-oo A noo n00...
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)) un...
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...
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the in...
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists M> 0 such that n(x) M for all r E [0,1] and all n N. (b) Does the result in part (a) hold if uniform convergence is replaced by pointwise convergence? Prove or give a counterexample
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists...
Proof Theorem 65.6 (a generalization of Dini's theorem) Let {fn be a sequence of real-valued continuous...
Proof
Theorem 65.6 (a generalization of Dini's theorem) Let {fn be a sequence of real-valued continuous functions on a compact subset S of R such that (1) for each x € S, the sequenсe {fn(x)}o is bounded and топotone, and (ii) the function x lim,0 fn(x) is continuous on S Then f Remark that the result is not always true without the monotonicity of item (i) Šn=0 lim fn uniformly on S
Theorem 65.6 (a generalization of Dini's theorem) Let...
(1) Let fn} be a sequence of C1 functions from the interval 0, 1] to R such that for some M 0 there holds fn(0) |(x)|&l...
(1) Let fn} be a sequence of C1 functions from the interval 0, 1] to R such that for some M 0 there holds fn(0) |(x)|< M, for al E [0, 1], nEN. Prove that there is a subsequence {fng} converging uniformly to a function f e C((0, 1]) on 0, 1].
(1) Let fn} be a sequence of C1 functions from the interval 0, 1] to R such that for some M 0 there holds fn(0) |(x)|
(5) Let {fn} be a sequence in C((0, 1]) which converges uniformly (to C([0, 1]). Prove that {fn} is uniformly bounded 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
(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
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
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 (, A, ) be a measure space. Let fn : 2 -» R* be a sequence of measurable functions. Let g,h : 2 -» R* be a pair of measurable functions such that both are integrable on that a set A E A and g(r) fn(x)h(x), for all E A and nE N. Prove / fn du lim sup A lim inf fn dulim inf lim sup fn du A fn du no0 no0 A noo n+o0 (You may...
(6) Let (, A,i) be a measure space. Let fn : 0 -» R* be a sequence of measurable functions. Let g, h : O -> R* be a pair of measurable functions such that both are integrable on a set A E A and g(x) < fn(x)<h(x), for all E A and ne N. Prove that / lim sup fn du fn dulim sup fn du lim inf fn du lim inf n o0 A n-oo A noo n00...
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...
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists M> 0 such that n(x) M for all r E [0,1] and all n N. (b) Does the result in part (a) hold if uniform convergence is replaced by pointwise convergence? Prove or give a counterexample
(2) Let {fJ be a sequence of continuous, real-valued functions that converges uniformly on the interval [0,1 (a) Show that there exists...
Proof
Theorem 65.6 (a generalization of Dini's theorem) Let {fn be a sequence of real-valued continuous functions on a compact subset S of R such that (1) for each x € S, the sequenсe {fn(x)}o is bounded and топotone, and (ii) the function x lim,0 fn(x) is continuous on S Then f Remark that the result is not always true without the monotonicity of item (i) Šn=0 lim fn uniformly on S
Theorem 65.6 (a generalization of Dini's theorem) Let...
(1) Let fn} be a sequence of C1 functions from the interval 0, 1] to R such that for some M 0 there holds fn(0) |(x)|< M, for al E [0, 1], nEN. Prove that there is a subsequence {fng} converging uniformly to a function f e C((0, 1]) on 0, 1].
(1) Let fn} be a sequence of C1 functions from the interval 0, 1] to R such that for some M 0 there holds fn(0) |(x)|
(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
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