Mixing
I don't understand the proof of theorem 7.25 in Rudin
$begingroup$
if $K$ is compact, $f_{n}$ is a complex continuous function on defined on $K$ for $n = 1,2,3,...$
and if $f_n$ is pointwise bounded and equicontinuous on $K$, then.
$f_n$ is uniformly bounded on $K$.
I don't understand, in the proof, why since $K$ is compact, there are finitely many points $p_1, ..., p_r$ in $K$ such that to every $x in K$ corresponds at least one $p_i$ with $d(x,p_i) < delta$.
real-analysis
edited Feb 1 at 14:31
Ernie060
2,940719
asked Mar 18 '15 at 2:47
john charliejohn charlie
61
$endgroup$
add a comment |
$begingroup$
if $K$ is compact, $f_{n}$ is a complex continuous function on defined on $K$ for $n = 1,2,3,...$
and if $f_n$ is pointwise bounded and equicontinuous on $K$, then.
$f_n$ is uniformly bounded on $K$.
I don't understand, in the proof, why since $K$ is compact, there are finitely many points $p_1, ..., p_r$ in $K$ such that to every $x in K$ corresponds at least one $p_i$ with $d(x,p_i) < delta$.
real-analysis
edited Feb 1 at 14:31
Ernie060
2,940719
asked Mar 18 '15 at 2:47
john charliejohn charlie
61
$endgroup$
add a comment |
$begingroup$
if $K$ is compact, $f_{n}$ is a complex continuous function on defined on $K$ for $n = 1,2,3,...$
and if $f_n$ is pointwise bounded and equicontinuous on $K$, then.
$f_n$ is uniformly bounded on $K$.
I don't understand, in the proof, why since $K$ is compact, there are finitely many points $p_1, ..., p_r$ in $K$ such that to every $x in K$ corresponds at least one $p_i$ with $d(x,p_i) < delta$.
real-analysis
edited Feb 1 at 14:31
Ernie060
2,940719
asked Mar 18 '15 at 2:47
john charliejohn charlie
61
$endgroup$
if $K$ is compact, $f_{n}$ is a complex continuous function on defined on $K$ for $n = 1,2,3,...$
and if $f_n$ is pointwise bounded and equicontinuous on $K$, then.
$f_n$ is uniformly bounded on $K$.
I don't understand, in the proof, why since $K$ is compact, there are finitely many points $p_1, ..., p_r$ in $K$ such that to every $x in K$ corresponds at least one $p_i$ with $d(x,p_i) < delta$.
real-analysis
real-analysis
edited Feb 1 at 14:31
Ernie060
2,940719
asked Mar 18 '15 at 2:47
john charliejohn charlie
61
edited Feb 1 at 14:31
Ernie060
2,940719
asked Mar 18 '15 at 2:47
john charliejohn charlie
61
edited Feb 1 at 14:31
Ernie060
2,940719
edited Feb 1 at 14:31
Ernie060
2,940719
edited Feb 1 at 14:31
Ernie060
2,940719
2,940719
asked Mar 18 '15 at 2:47
john charliejohn charlie
61
asked Mar 18 '15 at 2:47
john charliejohn charlie
61
asked Mar 18 '15 at 2:47
john charliejohn charlie
61
61
add a comment |
add a comment |
1 Answer
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$begingroup$
Consider the open cover of $K$ given by ${N_delta(k)}_{kin K}$. Apply compactness to find a finite subcover.
answered Mar 18 '15 at 3:01
zibadawa timmyzibadawa timmy
3,5661024
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
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$begingroup$
Consider the open cover of $K$ given by ${N_delta(k)}_{kin K}$. Apply compactness to find a finite subcover.
answered Mar 18 '15 at 3:01
zibadawa timmyzibadawa timmy
3,5661024
$endgroup$
add a comment |
$begingroup$
Consider the open cover of $K$ given by ${N_delta(k)}_{kin K}$. Apply compactness to find a finite subcover.
answered Mar 18 '15 at 3:01
zibadawa timmyzibadawa timmy
3,5661024
$endgroup$
add a comment |
$begingroup$
Consider the open cover of $K$ given by ${N_delta(k)}_{kin K}$. Apply compactness to find a finite subcover.
answered Mar 18 '15 at 3:01
zibadawa timmyzibadawa timmy
3,5661024
$endgroup$
Consider the open cover of $K$ given by ${N_delta(k)}_{kin K}$. Apply compactness to find a finite subcover.
answered Mar 18 '15 at 3:01
zibadawa timmyzibadawa timmy
3,5661024
answered Mar 18 '15 at 3:01
zibadawa timmyzibadawa timmy
3,5661024
answered Mar 18 '15 at 3:01
zibadawa timmyzibadawa timmy
3,5661024
answered Mar 18 '15 at 3:01
zibadawa timmyzibadawa timmy
3,5661024
3,5661024
add a comment |
add a comment |
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