So you need to proof x•c < x for 0<=c<1?
Isn’t that just:
xc < x | ÷x
c < x/x (for x=/=0)
c < 1 q.e.d.
What am I missing?
My math teacher would be angry because you started from the conclusion and derived the premise, rather than the other way around. Note also that you assumed that division is defined. That may not have been the case in the original problem.
Proof by contrapositive would be c<0 ∨ c≥1 ⇒ … ⇒ xc≥x. That is not just starting from the conclusion and deriving the premise.
Your math teacher is weird. But you can just turn it around:
c < 1
c < x/x | •x
xc < x q.e.d.
This also shows, that c≥0 is not actually a requirement, but x>0 is
I guess if your math teacher is completely insufferable, you need to add the definitions of the arithmetic operations but at that point you should also need to introduce Latin letters and Arabic numerals.