Fixed Point Theorem and Closed Unit Ball in Euclidean Space

The Brouwer Fixed-Point Theorem

Let   denote the closed unit ball in Euclidean  space  :
.
Any continuous map   from   onto itself has at least one fixed point, i.e. a point   such that  .

Proof Suppose   has no fixed points, i.e.   for  .  
Define a map   ,  , by letting   be the point of intersection of   and the ray starting at the point   and going through  .  For   see figure below:

We have
    with    ,     (1)
Then
   (2)
and so   is continuous.  Could you please explain, in as much detail as possible, how (1) and (2) were derived?

Attached file(s):

Brouwer Fixed-Point Theorem 1.doc  View File