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Math Help - Inequality proof

  1. #1
    MHF Contributor
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    Inequality proof

    \lVert \mathbf{u} + \mathbf{v} \rVert \geq  \lVert \mathbf{u} \rVert - \lVert \mathbf{v} \rVert

    \big(\lVert \mathbf{u} + \mathbf{v} \rVert \big)^2= <\mathbf{u} + \mathbf{v},\mathbf{u} + \mathbf{v}> = <\mathbf{v},\mathbf{v}> + 2<\mathbf{v},\mathbf{u}> + <\mathbf{u},\mathbf{u}>

    Cauchy-Schwarz: <\mathbf{v},\mathbf{u}> \leq  \lVert \mathbf{u} \rVert \lVert \mathbf{v} \rVert

    \lVert \mathbf{u} \rVert^2 + \lVert \mathbf{v} \rVert^2 + 2  \lVert \mathbf{u} \rVert \lVert \mathbf{v} \rVert

    Not sure I am getting anywhere
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  2. #2
    Senior Member
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    Use the triangle inequality. |y|=|y+x+(-x)| \le |y+x| + |x|, which is what you want.
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