# Norm Inequality Proof

• Sep 2nd 2008, 07:23 PM
Paperwings
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 $\| v \| < 1$
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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.
• Sep 2nd 2008, 08:26 PM
NonCommAlg
Quote:

Originally Posted by Paperwings
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: $||v|| = ||v-u + u|| \leq ||v-u|| + ||u|| < 1 - ||u|\ + ||u|| =1. \ \ \ \square$