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Critique my proof of the Arzela-Ascoli Theorem (with one question)

  1. Jan 16, 2012 #1
    In part (b), I'm not sure why it's important that r_1, r_2, ..., r_n is finite. Any thoughts? One thing I'm concerned about is if I have really shown that g_k converges uniformly because I did not shown that the N chosen was independent of the x.

    arzela1.png

    arzela2.png

    arzela3.png
     
  2. jcsd
  3. Jan 16, 2012 #2

    micromass

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    The problem is of course that every [itex]r_i[/itex] gives rise to a different N.

    That is: you know that [itex](g_s(r_i))_s[/itex] is Cauchy, thus we can write:

    [tex]\forall \varepsilon >0:\exists N_i: \forall s,t>N_i:~|g_s(r_i)-g_t(r_i)|<\varepsilon[/tex]

    I wrote [itex]N_i[/itex] here instead of [itex]N[/itex] because we do not have only one N.

    Now you must combine the [itex]N_i[/itex] into one N. This will use finiteness.


    A critique of your proof: you have not shown that we can actually choose a finite set [itex]\{r_1,...,r_n\}[/itex] that satisfies the criteria. This is very important and uses something essential.

    Furthermore, your proof of (c) isn't quite nice. You say "we may choose [itex]\delta>0[/itex]", but you do realize that this delta was already chosen in (b)?
     
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