I understand the steps of the proof in the book, but I don't see intuitively the of maximum increase at a point $P$ must be given by the $||\nabla f(x, y)||$. A graph has infinite directional derivatives at point $P$, I just don't see what is special about the sum of the directional derivatives in the $x$ direction and in the $y$ direction. If at a point $P$ the absolute value of the gradient is equal to the maximum increase, can't we just rotate the graph any amount around point $P$, the same maximum increase but a different gradient?
P.S. Please try to avoid using many advanced logic symbols.