Problem:

Let u be a point in $\displaystyle \mathbb{R}^n$ and suppose that $\displaystyle \| u \| < 1$. Show that ifvis in $\displaystyle \mathbb{R}^n$ and $\displaystyle \| v - u \| < 1 - \| u \|$, then $\displaystyle \| v \| < 1$

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Attempt

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

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

Square both sides, we get

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

Solve for $\displaystyle \| v \|^2 $

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

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

Cancelling out liked terms

$\displaystyle \| v \| ^2 < 1 - 2 \| u\| + 2 \left\langle u, v \right\rangle$ *

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

Thank you for your time and help.