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Help needed understanding Iterated Limits $lim_nrightarrow infty(lim_m rightarrow infty a_mn)$ and …
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Let $a_m,n= fracmm+n.$
Compute the iterated limits $$lim_nrightarrow infty(lim_m rightarrow infty a_mn)$$ and $$lim_mrightarrow infty(lim_n rightarrow infty a_mn)$$
My attempt at solving the first limit is that if we take $(lim_m rightarrow infty fracmm+n)$, for a fixed value of $n$ it should converge to $1$. And then $lim_nrightarrow infty (1)=1.$ And similary for the the second limit if we take $(lim_n rightarrow infty fracmm+n)$ for a fixed value of m, it converges to $0$ and then $lim_mrightarrow infty(0)=0$.
Query 1: I haven't a good understanding of iterated limits as this is the first time I'm encountering the topic. Is my reasoning correct?
Query 2: Also Stephen Abbott doesn't cover iterated limits in his book but gives it as a difficult last question to a chapter (I'm guessing I'm supposed to struggle with this a while) so are there any other sources/book out there that cover iterated limits for sequences (not integrals)?A simple google search does not help.
real-analysis sequences-and-series limits reference-request online-resources
asked Jul 23 at 23:37
Red
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up vote
2
down vote
favorite
Let $a_m,n= fracmm+n.$
Compute the iterated limits $$lim_nrightarrow infty(lim_m rightarrow infty a_mn)$$ and $$lim_mrightarrow infty(lim_n rightarrow infty a_mn)$$
My attempt at solving the first limit is that if we take $(lim_m rightarrow infty fracmm+n)$, for a fixed value of $n$ it should converge to $1$. And then $lim_nrightarrow infty (1)=1.$ And similary for the the second limit if we take $(lim_n rightarrow infty fracmm+n)$ for a fixed value of m, it converges to $0$ and then $lim_mrightarrow infty(0)=0$.
Query 1: I haven't a good understanding of iterated limits as this is the first time I'm encountering the topic. Is my reasoning correct?
Query 2: Also Stephen Abbott doesn't cover iterated limits in his book but gives it as a difficult last question to a chapter (I'm guessing I'm supposed to struggle with this a while) so are there any other sources/book out there that cover iterated limits for sequences (not integrals)?A simple google search does not help.
real-analysis sequences-and-series limits reference-request online-resources
asked Jul 23 at 23:37
Red
1,747733
You are correct.
– RRL
Jul 23 at 23:51
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $a_m,n= fracmm+n.$
Compute the iterated limits $$lim_nrightarrow infty(lim_m rightarrow infty a_mn)$$ and $$lim_mrightarrow infty(lim_n rightarrow infty a_mn)$$
My attempt at solving the first limit is that if we take $(lim_m rightarrow infty fracmm+n)$, for a fixed value of $n$ it should converge to $1$. And then $lim_nrightarrow infty (1)=1.$ And similary for the the second limit if we take $(lim_n rightarrow infty fracmm+n)$ for a fixed value of m, it converges to $0$ and then $lim_mrightarrow infty(0)=0$.
Query 1: I haven't a good understanding of iterated limits as this is the first time I'm encountering the topic. Is my reasoning correct?
Query 2: Also Stephen Abbott doesn't cover iterated limits in his book but gives it as a difficult last question to a chapter (I'm guessing I'm supposed to struggle with this a while) so are there any other sources/book out there that cover iterated limits for sequences (not integrals)?A simple google search does not help.
real-analysis sequences-and-series limits reference-request online-resources
asked Jul 23 at 23:37
Red
1,747733
Let $a_m,n= fracmm+n.$
Compute the iterated limits $$lim_nrightarrow infty(lim_m rightarrow infty a_mn)$$ and $$lim_mrightarrow infty(lim_n rightarrow infty a_mn)$$
My attempt at solving the first limit is that if we take $(lim_m rightarrow infty fracmm+n)$, for a fixed value of $n$ it should converge to $1$. And then $lim_nrightarrow infty (1)=1.$ And similary for the the second limit if we take $(lim_n rightarrow infty fracmm+n)$ for a fixed value of m, it converges to $0$ and then $lim_mrightarrow infty(0)=0$.
Query 1: I haven't a good understanding of iterated limits as this is the first time I'm encountering the topic. Is my reasoning correct?
Query 2: Also Stephen Abbott doesn't cover iterated limits in his book but gives it as a difficult last question to a chapter (I'm guessing I'm supposed to struggle with this a while) so are there any other sources/book out there that cover iterated limits for sequences (not integrals)?A simple google search does not help.
real-analysis sequences-and-series limits reference-request online-resources
asked Jul 23 at 23:37
Red
1,747733
asked Jul 23 at 23:37
Red
1,747733
asked Jul 23 at 23:37
Red
1,747733
asked Jul 23 at 23:37
Red
1,747733
1,747733
You are correct.
– RRL
Jul 23 at 23:51
add a comment |Â
You are correct.
– RRL
Jul 23 at 23:51
You are correct.
– RRL
Jul 23 at 23:51
You are correct.
– RRL
Jul 23 at 23:51
add a comment |Â
1 Answer
1
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oldest
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up vote
3
down vote
accepted
There is a general theory that considers iterated limits in relation to double limits and, among other things, explains under what conditions
$$tag*A=lim_n,m to infty a_mn = lim_mrightarrow infty(lim_n rightarrow infty a_mn) = lim_n.rightarrow infty(lim_m rightarrow infty a_mn)
$$
The general double limit is $A$ if for every $epsilon > 0$ there exists a positive integer $N$ such that if $n,m > N$, then $|a_mn - A| < epsilon$.
For example, if the double limit $lim_n,m to inftya_mn$ exists and each of the single limits $lim_n to infty a_mn$ and $lim_m to infty a_mn$ exist for all $m$ in the first case and all $n$ in the second case, then (*) holds.
A good reference is The Elements of Real Analysis (2nd edition) by Bartle.
answered Jul 23 at 23:51
RRL
43.6k42260
See The Moore-Osgood Theorem.
– Mark Viola
Jul 24 at 2:41
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
There is a general theory that considers iterated limits in relation to double limits and, among other things, explains under what conditions
$$tag*A=lim_n,m to infty a_mn = lim_mrightarrow infty(lim_n rightarrow infty a_mn) = lim_n.rightarrow infty(lim_m rightarrow infty a_mn)
$$
The general double limit is $A$ if for every $epsilon > 0$ there exists a positive integer $N$ such that if $n,m > N$, then $|a_mn - A| < epsilon$.
For example, if the double limit $lim_n,m to inftya_mn$ exists and each of the single limits $lim_n to infty a_mn$ and $lim_m to infty a_mn$ exist for all $m$ in the first case and all $n$ in the second case, then (*) holds.
A good reference is The Elements of Real Analysis (2nd edition) by Bartle.
answered Jul 23 at 23:51
RRL
43.6k42260
See The Moore-Osgood Theorem.
– Mark Viola
Jul 24 at 2:41
add a comment |Â
up vote
3
down vote
accepted
There is a general theory that considers iterated limits in relation to double limits and, among other things, explains under what conditions
$$tag*A=lim_n,m to infty a_mn = lim_mrightarrow infty(lim_n rightarrow infty a_mn) = lim_n.rightarrow infty(lim_m rightarrow infty a_mn)
$$
The general double limit is $A$ if for every $epsilon > 0$ there exists a positive integer $N$ such that if $n,m > N$, then $|a_mn - A| < epsilon$.
For example, if the double limit $lim_n,m to inftya_mn$ exists and each of the single limits $lim_n to infty a_mn$ and $lim_m to infty a_mn$ exist for all $m$ in the first case and all $n$ in the second case, then (*) holds.
A good reference is The Elements of Real Analysis (2nd edition) by Bartle.
answered Jul 23 at 23:51
RRL
43.6k42260
See The Moore-Osgood Theorem.
– Mark Viola
Jul 24 at 2:41
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
There is a general theory that considers iterated limits in relation to double limits and, among other things, explains under what conditions
$$tag*A=lim_n,m to infty a_mn = lim_mrightarrow infty(lim_n rightarrow infty a_mn) = lim_n.rightarrow infty(lim_m rightarrow infty a_mn)
$$
The general double limit is $A$ if for every $epsilon > 0$ there exists a positive integer $N$ such that if $n,m > N$, then $|a_mn - A| < epsilon$.
For example, if the double limit $lim_n,m to inftya_mn$ exists and each of the single limits $lim_n to infty a_mn$ and $lim_m to infty a_mn$ exist for all $m$ in the first case and all $n$ in the second case, then (*) holds.
A good reference is The Elements of Real Analysis (2nd edition) by Bartle.
answered Jul 23 at 23:51
RRL
43.6k42260
There is a general theory that considers iterated limits in relation to double limits and, among other things, explains under what conditions
$$tag*A=lim_n,m to infty a_mn = lim_mrightarrow infty(lim_n rightarrow infty a_mn) = lim_n.rightarrow infty(lim_m rightarrow infty a_mn)
$$
The general double limit is $A$ if for every $epsilon > 0$ there exists a positive integer $N$ such that if $n,m > N$, then $|a_mn - A| < epsilon$.
For example, if the double limit $lim_n,m to inftya_mn$ exists and each of the single limits $lim_n to infty a_mn$ and $lim_m to infty a_mn$ exist for all $m$ in the first case and all $n$ in the second case, then (*) holds.
A good reference is The Elements of Real Analysis (2nd edition) by Bartle.
answered Jul 23 at 23:51
RRL
43.6k42260
edited Jul 24 at 1:46
answered Jul 23 at 23:51
RRL
43.6k42260
answered Jul 23 at 23:51
RRL
43.6k42260
answered Jul 23 at 23:51
RRL
43.6k42260
43.6k42260
See The Moore-Osgood Theorem.
– Mark Viola
Jul 24 at 2:41
add a comment |Â
See The Moore-Osgood Theorem.
– Mark Viola
Jul 24 at 2:41
See The Moore-Osgood Theorem.
– Mark Viola
Jul 24 at 2:41
See The Moore-Osgood Theorem.
– Mark Viola
Jul 24 at 2:41
add a comment |Â
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You are correct.
– RRL
Jul 23 at 23:51