Induction and loop invariants

To hand in on paper in class:

Rosen, Section 4.1, pp. 280-82: Exercises 10 and 56
   Section 4.2, p. 292, Exercise 8
   Section 4.3, p. 309, Exercises 26ab

Find a loop invariant for the following subroutine and use it to prove that the subroutine is correct:

sub find_max {
   my(@nums) = @_;
   my($max) = $nums[0];
   for (my($i)=1; $i<@nums; $i++) {
      if ($nums[$i]>$max) { $max = $nums[$i]; }
   }
   return $max;
}

The assignment is worth 100 points.