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Math Help - Norm Inequality Proof

  1. #1
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    Norm Inequality Proof

    Problem:
    Let u be a point in \mathbb{R}^n and suppose that  \| u \| < 1. Show that if v is in \mathbb{R}^n and  \| v - u \| < 1 - \| u \|, then \| v \| < 1
    =================================

    Attempt

    Since \| v - u \|^2 = \| u\|^2 + \| v\|^2 - 2 \left\langle u, v \right\rangle, then

     \| v - u \| = \sqrt{\| u \|^2 + \|v \|^2 - 2 \left\langle u, v \right\rangle} < 1 - \| u \|

    Square both sides, we get

    \| u \|^2 + \| v \|^2 - 2 \left\langle u, v \right\rangle < \left( 1 - \| u\| \right)^2

    Solve for  \| v \|^2

    \| v \| ^2 < \left(1- \| u \| \right)^2 - \| u \|^2 + 2 \left\langle u, v \right\rangle

    \implies \| v \| ^2  <  1 - 2 \| u\| + \| u \|^2 - \| u \|^2 + 2 \left\langle u, v \right\rangle

    Cancelling out liked terms
     \| v \| ^2  <  1 - 2 \| u\| + 2 \left\langle u, v \right\rangle *

    From here, I am stuck. I am given that  \| u \| < 1. How would I prove that  \|  v \| < 1 from *?

    Thank you for your time and help.
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  2. #2
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    Quote Originally Posted by Paperwings View Post
    Problem:
    Let u be a point in \mathbb{R}^n and suppose that  \| u \| < 1. Show that if v is in \mathbb{R}^n and  \| v - u \| < 1 - \| u \|, then \| v \| < 1
    we need to have ||u|| < 1 for only one reason: because otherwise ||v-u|| < 1-||u||\leq 0, which is impossible. anyway, the solution

    to your problem is very simple: just use the triangle inequality: ||v|| = ||v-u + u|| \leq ||v-u|| + ||u|| < 1 - ||u|\ + ||u|| =1. \ \ \ \square
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