Suppose we have $$ points $$ in $$ in general position, with class labels $$ in $\{1,1\}.$

Prove that the perceptron learning algorithm converges to a separating hyperplane in a

finite number of steps:

a$.$ Denote a hyperplane by $()=1$

$+0=0,$ or in more compact notation

$=0,$ where $=(,1)$ and $=(1,0).$ Let $=$

$/||$

$||.$ Show that

separability implies the existence of a $$ such that $$

$>=1$

b$.$ Given a current $,$ the perceptron algorithm identifies a point $$ that is

misclassified and produces the update $+.$ Show th