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Math Help - Cauchy Sequence Inequality

  1. #1
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    Cauchy Sequence Inequality

    I am to prove CS inequality by mathematical induction. Don't even know where to start. Any help?
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    Super Member malaygoel's Avatar
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    Quote Originally Posted by Nichelle14
    I am to prove CS inequality by mathematical induction. Don't even know where to start. Any help?
    tell me the CS inequality.
    I don't care much about names so I can't figure it out.
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  3. #3
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    Perhaps it's Cauchy-Schwartz?
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    yes. sorry you are right it is Cauchey-Schwartz.
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  5. #5
    TD!
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    By induction? Induction on what then?
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    Quote Originally Posted by TD!
    By induction? Induction on what then?
    The dimension on the vector space.

    As I understand it she IS NOT talking about \mathbb{R}^3 but any dimesnion.
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  7. #7
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    The proof I usually use for Cauchy-Schwartz, in the form \Vert u \Vert^2 \Vert v \Vert^2 \ge (u\cdot v)^2, is to consider the quadratic f(x) = \Vert u + xv \Vert^2 \ge 0 and observe that the requirement ("b^2-4ac") that f has at most one real root is precisely C-S. This doesn't depend on dimension.

    If I had to do it by induction I think I'ld start by observing that it is trivial for n=1. The case n=2 follows from (x^2+y^2)(u^2+v^2) = (xu+yv)^2 + (xv-yu)^2 \ge (xu+yv)^2. The induction step would be something like ((u_1^2+\cdots+u_{n-1}^2)+u_n^2)((v_1+\cdots+v_{n-1}^2)+v_n^2) \ge \big\vert\sqrt{u_1^2+\cdots+u_{n-1}^2}\sqrt{v_1+\cdots+v_{n-1}^2}+u_nv_n \big\vert \ge \big\vert(u_1,\ldots,n_{n-1})\cdot(v_1,\ldots,v_{n-1})+u_nv_n\big\vert , thus using the cases 2 and n-1.
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  8. #8
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    I understand the problem as,
    for any n,
    |x_1y_1+...+x_ny_n|\leq \sqrt{x_1^2+...+x_n^2}\cdot \sqrt{y_1^2+...+y_n^2}
    For any x_i.y_i\in \mathbb{R}
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