Rudin's 'Principle of Mathematical Analysis' Problem 7.12 [closed]
$begingroup$
Suppose $g$ and $f_n$ ($n = 1,2,ldots$) are defined on $(0,infty)$, are Riemann-integrable on $[t,T]$ whenever $0 < t < T < infty$, $|f_n| leq g$, $f_n rightarrow f$ uniformly on every compact subset of $(0,infty)$, and
$$ int_0^{infty},g(x),dx < infty $$
Prove that
$$ lim_{n rightarrow infty},int_0^{infty},f_n(x),dx = int_0^{infty},f(x),dx $$
I appreciate all your comments, thanks.
sequences-and-series functional-analysis uniform-convergence
asked May 19 '14 at 17:52
Alexei0709Alexei0709
642411
$endgroup$
closed as off-topic by Eevee Trainer, Holo, KReiser, metamorphy, user91500 Dec 29 '18 at 10:26
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Eevee Trainer, Holo, KReiser, metamorphy, user91500
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Suppose $g$ and $f_n$ ($n = 1,2,ldots$) are defined on $(0,infty)$, are Riemann-integrable on $[t,T]$ whenever $0 < t < T < infty$, $|f_n| leq g$, $f_n rightarrow f$ uniformly on every compact subset of $(0,infty)$, and
$$ int_0^{infty},g(x),dx < infty $$
Prove that
$$ lim_{n rightarrow infty},int_0^{infty},f_n(x),dx = int_0^{infty},f(x),dx $$
I appreciate all your comments, thanks.
sequences-and-series functional-analysis uniform-convergence
asked May 19 '14 at 17:52
Alexei0709Alexei0709
642411
$endgroup$
closed as off-topic by Eevee Trainer, Holo, KReiser, metamorphy, user91500 Dec 29 '18 at 10:26
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Eevee Trainer, Holo, KReiser, metamorphy, user91500
If this question can be reworded to fit the rules in the help center, please edit the question.
4
$begingroup$
Show that $int_0^infty f_n(x),dx$ and $int_0^infty f(x),dx$ exist, and show the convergence by splitting the domain into two parts. One where you get a small difference from the domination by $g$, and one where you use the compact convergence.
$endgroup$
– Daniel Fischer♦
May 19 '14 at 17:59
add a comment |
$begingroup$
Suppose $g$ and $f_n$ ($n = 1,2,ldots$) are defined on $(0,infty)$, are Riemann-integrable on $[t,T]$ whenever $0 < t < T < infty$, $|f_n| leq g$, $f_n rightarrow f$ uniformly on every compact subset of $(0,infty)$, and
$$ int_0^{infty},g(x),dx < infty $$
Prove that
$$ lim_{n rightarrow infty},int_0^{infty},f_n(x),dx = int_0^{infty},f(x),dx $$
I appreciate all your comments, thanks.
sequences-and-series functional-analysis uniform-convergence
asked May 19 '14 at 17:52
Alexei0709Alexei0709
642411
$endgroup$
Suppose $g$ and $f_n$ ($n = 1,2,ldots$) are defined on $(0,infty)$, are Riemann-integrable on $[t,T]$ whenever $0 < t < T < infty$, $|f_n| leq g$, $f_n rightarrow f$ uniformly on every compact subset of $(0,infty)$, and
$$ int_0^{infty},g(x),dx < infty $$
Prove that
$$ lim_{n rightarrow infty},int_0^{infty},f_n(x),dx = int_0^{infty},f(x),dx $$
I appreciate all your comments, thanks.
sequences-and-series functional-analysis uniform-convergence
sequences-and-series functional-analysis uniform-convergence
asked May 19 '14 at 17:52
Alexei0709Alexei0709
642411
asked May 19 '14 at 17:52
Alexei0709Alexei0709
642411
edited May 19 '14 at 21:29
Alexei0709
asked May 19 '14 at 17:52
Alexei0709Alexei0709
642411
asked May 19 '14 at 17:52
Alexei0709Alexei0709
642411
asked May 19 '14 at 17:52
Alexei0709Alexei0709
642411
642411
closed as off-topic by Eevee Trainer, Holo, KReiser, metamorphy, user91500 Dec 29 '18 at 10:26
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Eevee Trainer, Holo, KReiser, metamorphy, user91500
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Eevee Trainer, Holo, KReiser, metamorphy, user91500 Dec 29 '18 at 10:26
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Eevee Trainer, Holo, KReiser, metamorphy, user91500
If this question can be reworded to fit the rules in the help center, please edit the question.
4
$begingroup$
Show that $int_0^infty f_n(x),dx$ and $int_0^infty f(x),dx$ exist, and show the convergence by splitting the domain into two parts. One where you get a small difference from the domination by $g$, and one where you use the compact convergence.
$endgroup$
– Daniel Fischer♦
May 19 '14 at 17:59
add a comment |
4
$begingroup$
Show that $int_0^infty f_n(x),dx$ and $int_0^infty f(x),dx$ exist, and show the convergence by splitting the domain into two parts. One where you get a small difference from the domination by $g$, and one where you use the compact convergence.
$endgroup$
– Daniel Fischer♦
May 19 '14 at 17:59
4
4
$begingroup$
Show that $int_0^infty f_n(x),dx$ and $int_0^infty f(x),dx$ exist, and show the convergence by splitting the domain into two parts. One where you get a small difference from the domination by $g$, and one where you use the compact convergence.
$endgroup$
– Daniel Fischer♦
May 19 '14 at 17:59
$begingroup$
Show that $int_0^infty f_n(x),dx$ and $int_0^infty f(x),dx$ exist, and show the convergence by splitting the domain into two parts. One where you get a small difference from the domination by $g$, and one where you use the compact convergence.
$endgroup$
– Daniel Fischer♦
May 19 '14 at 17:59
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Rather than writing my solution on here, I think its best if I gave you the official answer from the solution manual.
The solution manual of "Walter Rudin's Principles of Mathematical Analysis" can be found here
http://minds.wisconsin.edu/handle/1793/67009
I have also posted the answer to your question as an image below.
The images above contain the solution. Let me know if you need further help.
answered May 19 '14 at 22:43
meetbharathmeetbharath
544
$endgroup$
$begingroup$
Don't worry, this should be enough. Thank you so much!!!
$endgroup$
– Alexei0709
May 19 '14 at 22:57
$begingroup$
Your welcome :)
$endgroup$
– meetbharath
May 19 '14 at 23:00
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Rather than writing my solution on here, I think its best if I gave you the official answer from the solution manual.
The solution manual of "Walter Rudin's Principles of Mathematical Analysis" can be found here
http://minds.wisconsin.edu/handle/1793/67009
I have also posted the answer to your question as an image below.
The images above contain the solution. Let me know if you need further help.
answered May 19 '14 at 22:43
meetbharathmeetbharath
544
$endgroup$
$begingroup$
Don't worry, this should be enough. Thank you so much!!!
$endgroup$
– Alexei0709
May 19 '14 at 22:57
$begingroup$
Your welcome :)
$endgroup$
– meetbharath
May 19 '14 at 23:00
add a comment |
$begingroup$
Rather than writing my solution on here, I think its best if I gave you the official answer from the solution manual.
The solution manual of "Walter Rudin's Principles of Mathematical Analysis" can be found here
http://minds.wisconsin.edu/handle/1793/67009
I have also posted the answer to your question as an image below.
The images above contain the solution. Let me know if you need further help.
answered May 19 '14 at 22:43
meetbharathmeetbharath
544
$endgroup$
$begingroup$
Don't worry, this should be enough. Thank you so much!!!
$endgroup$
– Alexei0709
May 19 '14 at 22:57
$begingroup$
Your welcome :)
$endgroup$
– meetbharath
May 19 '14 at 23:00
add a comment |
$begingroup$
Rather than writing my solution on here, I think its best if I gave you the official answer from the solution manual.
The solution manual of "Walter Rudin's Principles of Mathematical Analysis" can be found here
http://minds.wisconsin.edu/handle/1793/67009
I have also posted the answer to your question as an image below.
The images above contain the solution. Let me know if you need further help.
answered May 19 '14 at 22:43
meetbharathmeetbharath
544
$endgroup$
Rather than writing my solution on here, I think its best if I gave you the official answer from the solution manual.
The solution manual of "Walter Rudin's Principles of Mathematical Analysis" can be found here
http://minds.wisconsin.edu/handle/1793/67009
I have also posted the answer to your question as an image below.
The images above contain the solution. Let me know if you need further help.
answered May 19 '14 at 22:43
meetbharathmeetbharath
544
answered May 19 '14 at 22:43
meetbharathmeetbharath
544
answered May 19 '14 at 22:43
meetbharathmeetbharath
544
answered May 19 '14 at 22:43
meetbharathmeetbharath
544
544
$begingroup$
Don't worry, this should be enough. Thank you so much!!!
$endgroup$
– Alexei0709
May 19 '14 at 22:57
$begingroup$
Your welcome :)
$endgroup$
– meetbharath
May 19 '14 at 23:00
add a comment |
$begingroup$
Don't worry, this should be enough. Thank you so much!!!
$endgroup$
– Alexei0709
May 19 '14 at 22:57
$begingroup$
Your welcome :)
$endgroup$
– meetbharath
May 19 '14 at 23:00
$begingroup$
Don't worry, this should be enough. Thank you so much!!!
$endgroup$
– Alexei0709
May 19 '14 at 22:57
$begingroup$
Don't worry, this should be enough. Thank you so much!!!
$endgroup$
– Alexei0709
May 19 '14 at 22:57
$begingroup$
Your welcome :)
$endgroup$
– meetbharath
May 19 '14 at 23:00
$begingroup$
Your welcome :)
$endgroup$
– meetbharath
May 19 '14 at 23:00
add a comment |
4
$begingroup$
Show that $int_0^infty f_n(x),dx$ and $int_0^infty f(x),dx$ exist, and show the convergence by splitting the domain into two parts. One where you get a small difference from the domination by $g$, and one where you use the compact convergence.
$endgroup$
– Daniel Fischer♦
May 19 '14 at 17:59