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Math Help - how to prove this..?

  1. #1
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    how to prove this..?

    D is subset of real number in p dimension
    [ (\vec{x_{n}}), n belongs natural number] is a subset of D
    \vec{x} belongs to D
    suppose (\vec{x_{n}}), n belongs natural number converges to \vec{x}
    f: D->R is cont' on D and f( (\vec{x_{n}}))<= r for all n belongs to natural number
    Prove that f( \vec{x})<=r

    i have totally no idea about wt is happening here.
    suppose (\vec{x_{n}}), n belongs natural number converges to \vec{x}
    this sentence means lim n->infinity (\vec{x_{n}}) = \vec{x}?
    and then that means lim n->infinity f( \vec{x_{n}}) = f( \vec{x}) ?
    how can these help me to derive that...thank you so much
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  2. #2
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    Oct 2007
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    London / Cambridge
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    Quote Originally Posted by pokemon1111 View Post
    D is subset of real number in p dimension
    [ (\vec{x_{n}}), n belongs natural number] is a subset of D
    \vec{x} belongs to D
    suppose (\vec{x_{n}}), n belongs natural number converges to \vec{x}
    f: D->R is cont' on D and f( (\vec{x_{n}}))<= r for all n belongs to natural number
    Prove that f( \vec{x})<=r

    i have totally no idea about wt is happening here.
    suppose (\vec{x_{n}}), n belongs natural number converges to \vec{x}
    this sentence means lim n->infinity (\vec{x_{n}}) = \vec{x}?
    and then that means lim n->infinity f( \vec{x_{n}}) = f( \vec{x}) ?
    how can these help me to derive that...thank you so much
    Have you tired a simple epsilon argument ?

    Bobak
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  3. #3
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    Quote Originally Posted by bobak View Post
    Have you tired a simple epsilon argument ?

    Bobak
    what does it mean?
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  4. #4
    Super Member
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    Oct 2007
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    London / Cambridge
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    Quote Originally Posted by pokemon1111 View Post
    what does it mean?
    An argument that uses the definition of limits and continuity.

    Bobak
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  5. #5
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    Quote Originally Posted by bobak View Post
    An argument that uses the definition of limits and continuity.

    Bobak
    || (\vec{x_{n}})- \vec{x}||< epsilon
    implies
    ||f( (\vec{x_{n}})) - f( \vec{x})||< delta
    ?
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