I'm studying about linear programming and I bumped into following theorem (I have added my questions into the image):
So in 1st rectangle how did we obtain the resulting equation when $\alpha \rightarrow 0_+$? Is it because $\alpha^2$ converges faster to zero than $\alpha$ and that's why we get the result, because the whole thing needs to be greater than zero?
In the 2nd case how does one conclude the geometric interpretation from the theorem? Example?
Thank you for any help :)
OK, I'll copy my comments as an answer not to leave the question in the unanswered section.
Just as the OP says. Divide both sides by $2\alpha$ and you will get that the scalar product $(x_0-y)^T(x-x_0)$ plus something tending to zero is nonnegative, which means that the scalar product is nonnegative.
The hyperplane is given by the formula $H=\{x\ | \ a^Tx=a^Ty\}$. It divides the space into two halves: where $a^Tx>a^Ty$ and $a^Tx<a^Ty$ respectively. The set C is of course in the first half.
P.S. $H$ is a hyperplane by the rank theorem: it is the kernel of matrix $a^T$ translated by vector $y$.