Is a sequence of limits bounded
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If you take an infinite sequence of infinite sequences and then find that the first element in every sequence converges to a point (because it is Cauchy and in a complete space), and then the second element in every sequence converges...and so on. If you take the sequence of those limit points, is it bounded?
EX) ${a_1, a_2, a_3,....};\
{b_1, b_2, b_3,....};\
{c_1, c_2, c_3,....};cdots$
Where ${a_1, b_1, c_1,....}$ is Cauchy and converges to a point, and ${a_2, b_2, c_2,....}$ is Cauchy and converges to a point, etc. Then how would I show that the sequence of the limit points is bounded. (Knowing that the sequence of sequences is Cauchy and bounded and the each column is also Cauchy and bounded).
real-analysis analysis
edited Nov 18 at 6:25
Yadati Kiran
1,243417
asked Nov 18 at 5:19
Bad at Math
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add a comment |
up vote
0
down vote
favorite
If you take an infinite sequence of infinite sequences and then find that the first element in every sequence converges to a point (because it is Cauchy and in a complete space), and then the second element in every sequence converges...and so on. If you take the sequence of those limit points, is it bounded?
EX) ${a_1, a_2, a_3,....};\
{b_1, b_2, b_3,....};\
{c_1, c_2, c_3,....};cdots$
Where ${a_1, b_1, c_1,....}$ is Cauchy and converges to a point, and ${a_2, b_2, c_2,....}$ is Cauchy and converges to a point, etc. Then how would I show that the sequence of the limit points is bounded. (Knowing that the sequence of sequences is Cauchy and bounded and the each column is also Cauchy and bounded).
real-analysis analysis
edited Nov 18 at 6:25
Yadati Kiran
1,243417
asked Nov 18 at 5:19
Bad at Math
1
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
If you take an infinite sequence of infinite sequences and then find that the first element in every sequence converges to a point (because it is Cauchy and in a complete space), and then the second element in every sequence converges...and so on. If you take the sequence of those limit points, is it bounded?
EX) ${a_1, a_2, a_3,....};\
{b_1, b_2, b_3,....};\
{c_1, c_2, c_3,....};cdots$
Where ${a_1, b_1, c_1,....}$ is Cauchy and converges to a point, and ${a_2, b_2, c_2,....}$ is Cauchy and converges to a point, etc. Then how would I show that the sequence of the limit points is bounded. (Knowing that the sequence of sequences is Cauchy and bounded and the each column is also Cauchy and bounded).
real-analysis analysis
edited Nov 18 at 6:25
Yadati Kiran
1,243417
asked Nov 18 at 5:19
Bad at Math
1
If you take an infinite sequence of infinite sequences and then find that the first element in every sequence converges to a point (because it is Cauchy and in a complete space), and then the second element in every sequence converges...and so on. If you take the sequence of those limit points, is it bounded?
EX) ${a_1, a_2, a_3,....};\
{b_1, b_2, b_3,....};\
{c_1, c_2, c_3,....};cdots$
Where ${a_1, b_1, c_1,....}$ is Cauchy and converges to a point, and ${a_2, b_2, c_2,....}$ is Cauchy and converges to a point, etc. Then how would I show that the sequence of the limit points is bounded. (Knowing that the sequence of sequences is Cauchy and bounded and the each column is also Cauchy and bounded).
real-analysis analysis
real-analysis analysis
edited Nov 18 at 6:25
Yadati Kiran
1,243417
asked Nov 18 at 5:19
Bad at Math
1
edited Nov 18 at 6:25
Yadati Kiran
1,243417
asked Nov 18 at 5:19
Bad at Math
1
edited Nov 18 at 6:25
Yadati Kiran
1,243417
edited Nov 18 at 6:25
Yadati Kiran
1,243417
edited Nov 18 at 6:25
Yadati Kiran
1,243417
1,243417
asked Nov 18 at 5:19
Bad at Math
1
asked Nov 18 at 5:19
Bad at Math
1
asked Nov 18 at 5:19
Bad at Math
1
1
add a comment |
add a comment |
1 Answer
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Suppose that the sequence of $n$-terms is the constant $n$ $a_1=1,b_=1,c_1=1; a_2=2,b_2=2,c_2=2,...$, each of these sequences converges converges but their limit points is not bounded.
answered Nov 18 at 5:22
Tsemo Aristide
54.6k11444
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1 Answer
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1 Answer
1
active
oldest
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active
oldest
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active
oldest
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up vote
0
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Suppose that the sequence of $n$-terms is the constant $n$ $a_1=1,b_=1,c_1=1; a_2=2,b_2=2,c_2=2,...$, each of these sequences converges converges but their limit points is not bounded.
answered Nov 18 at 5:22
Tsemo Aristide
54.6k11444
add a comment |
up vote
0
down vote
Suppose that the sequence of $n$-terms is the constant $n$ $a_1=1,b_=1,c_1=1; a_2=2,b_2=2,c_2=2,...$, each of these sequences converges converges but their limit points is not bounded.
answered Nov 18 at 5:22
Tsemo Aristide
54.6k11444
add a comment |
up vote
0
down vote
up vote
0
down vote
Suppose that the sequence of $n$-terms is the constant $n$ $a_1=1,b_=1,c_1=1; a_2=2,b_2=2,c_2=2,...$, each of these sequences converges converges but their limit points is not bounded.
answered Nov 18 at 5:22
Tsemo Aristide
54.6k11444
Suppose that the sequence of $n$-terms is the constant $n$ $a_1=1,b_=1,c_1=1; a_2=2,b_2=2,c_2=2,...$, each of these sequences converges converges but their limit points is not bounded.
answered Nov 18 at 5:22
Tsemo Aristide
54.6k11444
answered Nov 18 at 5:22
Tsemo Aristide
54.6k11444
answered Nov 18 at 5:22
Tsemo Aristide
54.6k11444
answered Nov 18 at 5:22
Tsemo Aristide
54.6k11444
54.6k11444
add a comment |
add a comment |
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