The weaker boundedness implies uniformly bounded












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Theorem (Arzelà-Ascoli). Eevery bounded equicontinuous sequence of functions in $C^{0}([a,b],mathbb{R})$ has a uniformly convergent subsequence.




The question asks to generalize the theorem with the weaker boundedness hypothesis. So, the theorem becomes




Theorem (Arzelà-Ascoli generalization). Every equicontinuous sequence $(f_{n})$ of functions with compact domain such that for $x in K$, $(f_{n}(x))$ is bounded in $C^{0}(K,mathbb{R})$ has a uniformly convergent subsequence.




So, I want to prove that




$$text{If $(f_{n}(x))$ is a bounded sequence for $x in K$, then $(f_{n})$ is uniformly bounded.}$$




I didnt get a good idea to start. Could someone give me just a hint?






equicontinuity arzela-ascoli







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  • $begingroup$
    The proof is outlined here: en.wikipedia.org/wiki/…
    $endgroup$
    – Ben W
    Jan 10 at 2:31
















0












$begingroup$



Theorem (Arzelà-Ascoli). Eevery bounded equicontinuous sequence of functions in $C^{0}([a,b],mathbb{R})$ has a uniformly convergent subsequence.




The question asks to generalize the theorem with the weaker boundedness hypothesis. So, the theorem becomes




Theorem (Arzelà-Ascoli generalization). Every equicontinuous sequence $(f_{n})$ of functions with compact domain such that for $x in K$, $(f_{n}(x))$ is bounded in $C^{0}(K,mathbb{R})$ has a uniformly convergent subsequence.




So, I want to prove that




$$text{If $(f_{n}(x))$ is a bounded sequence for $x in K$, then $(f_{n})$ is uniformly bounded.}$$




I didnt get a good idea to start. Could someone give me just a hint?






equicontinuity arzela-ascoli







share|cite|improve this question









$endgroup$












  • $begingroup$
    The proof is outlined here: en.wikipedia.org/wiki/…
    $endgroup$
    – Ben W
    Jan 10 at 2:31














0












0








0





$begingroup$



Theorem (Arzelà-Ascoli). Eevery bounded equicontinuous sequence of functions in $C^{0}([a,b],mathbb{R})$ has a uniformly convergent subsequence.




The question asks to generalize the theorem with the weaker boundedness hypothesis. So, the theorem becomes




Theorem (Arzelà-Ascoli generalization). Every equicontinuous sequence $(f_{n})$ of functions with compact domain such that for $x in K$, $(f_{n}(x))$ is bounded in $C^{0}(K,mathbb{R})$ has a uniformly convergent subsequence.




So, I want to prove that




$$text{If $(f_{n}(x))$ is a bounded sequence for $x in K$, then $(f_{n})$ is uniformly bounded.}$$




I didnt get a good idea to start. Could someone give me just a hint?






equicontinuity arzela-ascoli







share|cite|improve this question









$endgroup$





Theorem (Arzelà-Ascoli). Eevery bounded equicontinuous sequence of functions in $C^{0}([a,b],mathbb{R})$ has a uniformly convergent subsequence.




The question asks to generalize the theorem with the weaker boundedness hypothesis. So, the theorem becomes




Theorem (Arzelà-Ascoli generalization). Every equicontinuous sequence $(f_{n})$ of functions with compact domain such that for $x in K$, $(f_{n}(x))$ is bounded in $C^{0}(K,mathbb{R})$ has a uniformly convergent subsequence.




So, I want to prove that




$$text{If $(f_{n}(x))$ is a bounded sequence for $x in K$, then $(f_{n})$ is uniformly bounded.}$$




I didnt get a good idea to start. Could someone give me just a hint?






equicontinuity arzela-ascoli




equicontinuity arzela-ascoli






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




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asked Jan 10 at 2:24









Lucas CorrêaLucas Corrêa

1,6151321




1,6151321












  • $begingroup$
    The proof is outlined here: en.wikipedia.org/wiki/…
    $endgroup$
    – Ben W
    Jan 10 at 2:31


















  • $begingroup$
    The proof is outlined here: en.wikipedia.org/wiki/…
    $endgroup$
    – Ben W
    Jan 10 at 2:31
















$begingroup$
The proof is outlined here: en.wikipedia.org/wiki/…
$endgroup$
– Ben W
Jan 10 at 2:31




$begingroup$
The proof is outlined here: en.wikipedia.org/wiki/…
$endgroup$
– Ben W
Jan 10 at 2:31










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