deletions | additions       diff --git a/untitled.tex b/untitled.tex  index d4e8689..f6a47a6 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
1. - 1.71. Our proof of the Cauchy-Schwarz inequality, Theorem 1.13, used that when  $U$ is a unit vector,  $0 \leq ||V−(U·V)U||^2 = ||V||2 −(U·V)^2$. 
...
$$=x_1^2 + x_2^2+\dots+ x_n ^2$$  $$= ||X||^2$$  Since norms are nonnegative, we can conclude that $||CX|| = ||X||$.  3.