Sorry, the textbook is not clear. Let's collect all these bugs to make our lives easier.
The most "popular" mistake:
Step 1: try to make a scenario (usually setting s=0) and claim X(t)={A(t)-c*t} for any t > 0.
Step 2: try to prove that this is the worst case scenario.
I do
not think step 1 is correct. You may not find a scenario that X(t)={A(t)-c*t}. (e.g. A(t) < c(t) all the time. Then X(t) is always equal to zero).
Lesson:
1. You cannot "prove a general theorem" by a single scenario or some (uncomplete) set of scenarios
2. When you write "Let
q= sup S" or "
q = inf S", you have to be sure that S is not an empty set.