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Use of Arzela-Ascoli theorem to ensure that there is a subsequence of a sequence of functions that converges to an infinitely differentiable function.
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I'm trying understand how to prove this result. I read the answer given on the post, but there are three points of the answer that I didn't understand:
Why $f'_n_k_i rightarrow f'?$ I know that $f'_n_k_i rightarrow g$ for some function $g$ by Arzela-Ascoli's theorem, but how exactly can I ensure that $g = f'?$
How ensure that exists $f'?$ Because the answer only ensure that $f$ is continuous and I don't think that Mean Value Theorem can be ensure that $f in mathcalC^1$.
- Why is necessary to use the Cantor's diagonal argument? As the author of the answer said, is the process of choosing subsequences no longer enough to ensure the function's differentiability?
Thanks in advance!
general-topology proof-explanation arzela-ascoli
asked Jul 30 at 1:38
George
659512
add a comment |Â
up vote
0
down vote
favorite
I'm trying understand how to prove this result. I read the answer given on the post, but there are three points of the answer that I didn't understand:
Why $f'_n_k_i rightarrow f'?$ I know that $f'_n_k_i rightarrow g$ for some function $g$ by Arzela-Ascoli's theorem, but how exactly can I ensure that $g = f'?$
How ensure that exists $f'?$ Because the answer only ensure that $f$ is continuous and I don't think that Mean Value Theorem can be ensure that $f in mathcalC^1$.
- Why is necessary to use the Cantor's diagonal argument? As the author of the answer said, is the process of choosing subsequences no longer enough to ensure the function's differentiability?
Thanks in advance!
general-topology proof-explanation arzela-ascoli
asked Jul 30 at 1:38
George
659512
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm trying understand how to prove this result. I read the answer given on the post, but there are three points of the answer that I didn't understand:
Why $f'_n_k_i rightarrow f'?$ I know that $f'_n_k_i rightarrow g$ for some function $g$ by Arzela-Ascoli's theorem, but how exactly can I ensure that $g = f'?$
How ensure that exists $f'?$ Because the answer only ensure that $f$ is continuous and I don't think that Mean Value Theorem can be ensure that $f in mathcalC^1$.
- Why is necessary to use the Cantor's diagonal argument? As the author of the answer said, is the process of choosing subsequences no longer enough to ensure the function's differentiability?
Thanks in advance!
general-topology proof-explanation arzela-ascoli
asked Jul 30 at 1:38
George
659512
I'm trying understand how to prove this result. I read the answer given on the post, but there are three points of the answer that I didn't understand:
Why $f'_n_k_i rightarrow f'?$ I know that $f'_n_k_i rightarrow g$ for some function $g$ by Arzela-Ascoli's theorem, but how exactly can I ensure that $g = f'?$
How ensure that exists $f'?$ Because the answer only ensure that $f$ is continuous and I don't think that Mean Value Theorem can be ensure that $f in mathcalC^1$.
- Why is necessary to use the Cantor's diagonal argument? As the author of the answer said, is the process of choosing subsequences no longer enough to ensure the function's differentiability?
Thanks in advance!
general-topology proof-explanation arzela-ascoli
asked Jul 30 at 1:38
George
659512
asked Jul 30 at 1:38
George
659512
asked Jul 30 at 1:38
George
659512
asked Jul 30 at 1:38
George
659512
659512
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
Suppose $f_n to f$ uniformly and $f_n' to g$ uniformly. Then $f_n(x)-f_n(x_0)=int_x_0^x f_n'(t) , dt$. Take limits to get $f(x)-f(x_0)=int_x_0^x g(t), dt$. This implies that $f$ is differentiable and $f'=g$. Repeated use of this theorem answers all your questions.
answered Jul 30 at 6:39
Kavi Rama Murthy
19.7k2829
I understand how this answer my first two questions, but how this answer the third?
– George
Jul 30 at 13:27
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Suppose $f_n to f$ uniformly and $f_n' to g$ uniformly. Then $f_n(x)-f_n(x_0)=int_x_0^x f_n'(t) , dt$. Take limits to get $f(x)-f(x_0)=int_x_0^x g(t), dt$. This implies that $f$ is differentiable and $f'=g$. Repeated use of this theorem answers all your questions.
answered Jul 30 at 6:39
Kavi Rama Murthy
19.7k2829
I understand how this answer my first two questions, but how this answer the third?
– George
Jul 30 at 13:27
add a comment |Â
up vote
2
down vote
accepted
Suppose $f_n to f$ uniformly and $f_n' to g$ uniformly. Then $f_n(x)-f_n(x_0)=int_x_0^x f_n'(t) , dt$. Take limits to get $f(x)-f(x_0)=int_x_0^x g(t), dt$. This implies that $f$ is differentiable and $f'=g$. Repeated use of this theorem answers all your questions.
answered Jul 30 at 6:39
Kavi Rama Murthy
19.7k2829
I understand how this answer my first two questions, but how this answer the third?
– George
Jul 30 at 13:27
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Suppose $f_n to f$ uniformly and $f_n' to g$ uniformly. Then $f_n(x)-f_n(x_0)=int_x_0^x f_n'(t) , dt$. Take limits to get $f(x)-f(x_0)=int_x_0^x g(t), dt$. This implies that $f$ is differentiable and $f'=g$. Repeated use of this theorem answers all your questions.
answered Jul 30 at 6:39
Kavi Rama Murthy
19.7k2829
Suppose $f_n to f$ uniformly and $f_n' to g$ uniformly. Then $f_n(x)-f_n(x_0)=int_x_0^x f_n'(t) , dt$. Take limits to get $f(x)-f(x_0)=int_x_0^x g(t), dt$. This implies that $f$ is differentiable and $f'=g$. Repeated use of this theorem answers all your questions.
answered Jul 30 at 6:39
Kavi Rama Murthy
19.7k2829
edited Jul 30 at 8:40
answered Jul 30 at 6:39
Kavi Rama Murthy
19.7k2829
answered Jul 30 at 6:39
Kavi Rama Murthy
19.7k2829
answered Jul 30 at 6:39
Kavi Rama Murthy
19.7k2829
19.7k2829
I understand how this answer my first two questions, but how this answer the third?
– George
Jul 30 at 13:27
add a comment |Â
I understand how this answer my first two questions, but how this answer the third?
– George
Jul 30 at 13:27
I understand how this answer my first two questions, but how this answer the third?
– George
Jul 30 at 13:27
I understand how this answer my first two questions, but how this answer the third?
– George
Jul 30 at 13:27
add a comment |Â
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