Prove that a subset $A$ of real numbers is compact iff for any $B$ (infinite) subset of $A$ there exists some...
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My attempt:
Suppose ${a_n}$ is a sequence in $B$. Since $B$ is a subset of $A$ and $A$ is compact, there exists some ${a_{n_k}}$ subsequence of ${a_n}$ which converges to $x$. Now can I say $x$ is a limit point of $A$?
general-topology
edited Dec 21 '18 at 2:36
Saad
20.4k92452
asked Dec 18 '18 at 19:47
Arman_jrArman_jr
235
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add a comment |
$begingroup$
My attempt:
Suppose ${a_n}$ is a sequence in $B$. Since $B$ is a subset of $A$ and $A$ is compact, there exists some ${a_{n_k}}$ subsequence of ${a_n}$ which converges to $x$. Now can I say $x$ is a limit point of $A$?
general-topology
edited Dec 21 '18 at 2:36
Saad
20.4k92452
asked Dec 18 '18 at 19:47
Arman_jrArman_jr
235
$endgroup$
$begingroup$
Can you provide the definition of limit point that you are using?
$endgroup$
– Matheus Manzatto
Dec 21 '18 at 2:24
add a comment |
$begingroup$
My attempt:
Suppose ${a_n}$ is a sequence in $B$. Since $B$ is a subset of $A$ and $A$ is compact, there exists some ${a_{n_k}}$ subsequence of ${a_n}$ which converges to $x$. Now can I say $x$ is a limit point of $A$?
general-topology
edited Dec 21 '18 at 2:36
Saad
20.4k92452
asked Dec 18 '18 at 19:47
Arman_jrArman_jr
235
$endgroup$
My attempt:
Suppose ${a_n}$ is a sequence in $B$. Since $B$ is a subset of $A$ and $A$ is compact, there exists some ${a_{n_k}}$ subsequence of ${a_n}$ which converges to $x$. Now can I say $x$ is a limit point of $A$?
general-topology
general-topology
edited Dec 21 '18 at 2:36
Saad
20.4k92452
asked Dec 18 '18 at 19:47
Arman_jrArman_jr
235
edited Dec 21 '18 at 2:36
Saad
20.4k92452
asked Dec 18 '18 at 19:47
Arman_jrArman_jr
235
edited Dec 21 '18 at 2:36
Saad
20.4k92452
edited Dec 21 '18 at 2:36
Saad
20.4k92452
edited Dec 21 '18 at 2:36
Saad
20.4k92452
20.4k92452
asked Dec 18 '18 at 19:47
Arman_jrArman_jr
235
asked Dec 18 '18 at 19:47
Arman_jrArman_jr
235
asked Dec 18 '18 at 19:47
Arman_jrArman_jr
235
235
$begingroup$
Can you provide the definition of limit point that you are using?
$endgroup$
– Matheus Manzatto
Dec 21 '18 at 2:24
add a comment |
$begingroup$
Can you provide the definition of limit point that you are using?
$endgroup$
– Matheus Manzatto
Dec 21 '18 at 2:24
$begingroup$
Can you provide the definition of limit point that you are using?
$endgroup$
– Matheus Manzatto
Dec 21 '18 at 2:24
$begingroup$
Can you provide the definition of limit point that you are using?
$endgroup$
– Matheus Manzatto
Dec 21 '18 at 2:24
add a comment |
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$begingroup$
Can you provide the definition of limit point that you are using?
$endgroup$
– Matheus Manzatto
Dec 21 '18 at 2:24