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Let An Bn and Cn be sequences satisfying An<=Bn<=Cn

  1. Jun 16, 2005 #1
    hello all

    been workin on this problem:
    let An Bn and Cn be sequences satisfying
    An<=Bn<=Cn for all n an element of the natural numbers
    suppose that An->x and Cn->x, where x is a real number show that Bn->x
    this is how i did it


    [tex]A_n\le B_n\le C_n \forall n\epsilon N[/tex]

    [tex]A_n\longrightarrow x,C_n\longrightarrow x\ \forall x\epsilon \Re[/tex]

    [tex]\lim_{n\to\infty}A_n\le\lim_{n\to\infty}B_n\le\lim_{n\to\infty}C_n[/tex]

    [tex]x\le\lim_{n\to\infty}B_n\le x[/tex]

    therefore by the squeeze theorem [tex]B_n\longrightarrow x[/tex]

    would this be correct, and are there any other ways of proving it?

    thanxs
     
  2. jcsd
  3. Jun 16, 2005 #2

    quasar987

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    What you wrote it confusing, for many reasons. First, the question

    "Let An Bn and Cn be sequences satisfying An<=Bn<=Cn for all n an element of the natural numbers. Suppose that An->x and Cn->x, where x is a real number. Show that Bn->x."

    must mean "prove the squeeze theorem". Otherwise, Bn->x is just the statement of the squeeze theorem and there's nothing to show at all.

    Secondly, line #2 makes no sense (because if the limit of An exists, it is unique), but it was probably a typo.

    Thirdly, you invoque the squeeze theorem after line #4 tu justify that Bn->x. But this is just a consequence of the axiom of the real numbers according to which for all x,y in R, we can only have one of the 3: x<y, x=y, x>y. So if we encounter an inequality of the type [itex]y\leq x \leq y[/itex], it must be that y=x. So lim Bn = x.


    But in essence, your had the right proof.
     
  4. Jun 16, 2005 #3

    quasar987

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    The real rigourous proof involves the old N and [itex]\epsilon[/itex] though... because that's the language to use when one talks about limits.
     
  5. Jun 16, 2005 #4
    It seems to me that this proof assumes that [itex]\lim_{n\rightarrow\infty}B_n[/itex] exists.
     
  6. Jun 16, 2005 #5
    hello guys

    well I had proved it through N-E method in which was succesful, based upon your replies above, would i be right to say that my original method is not sufficiant enough to prove it since i have used the assumption that the limit of Bn exists and that i have used the sqeeze theorem while trying to prove the sqeeze theorem, which is i think they call a fallacy,

    steven
     
  7. Jun 17, 2005 #6

    quasar987

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    I already pointed out that what you did after line #4 is not used the squeeze theorem but simply the axioms of the real numbers.

    But I think master coda has a good point.. and in an exam that proof wouldn't be worth many points imo.
     
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