[Proof] can be tricky

So we finished the first midterm :) It was quit easy (hope I did not make any stupid mistakes)
After the midterm we started the lecture. Larry was funny as always he pulled out a proof which was:
(I will copy the proof down to review the basic structure of proof)
Proof:
assume you left all questions blank
#that's pretty bad!                                -----> remember to put comment when you right a proof
    then you get 20%
    assume class average is 70% #pretty high!
        then you are 50% below average
        in this term test #70% - 20% = 50%
        then this term test weighs 6% of final grade
        # according course info sheet
        then you are 3% below average
        in term of final grade #6%*50%=3%
        then it is below are the acceptable
        margin of error #5% in physics
        then it is totally acceptable
then it is not bad even if you left everything blank and others did well.
                                                                 ----> remember the last ''then'' will in line with first ''assume''


Last week, we learnt direct proof for universally quantified implication which is for all x belongs to X, P(x) => Q(x).

For example:

         

However, sometimes it is hard to prove this way so we might try prove the contrapositive of the original statement. 

Sometimes it is not clear what P is, in this situation, we use contradiction.
For Example, to prove "There are infinitely many even natural numbers." We suppose there is a finite number of even number => then there must be a largest one, call it X => but if I double X => I get a larger number, a larger even number => so X is NOT the largest one.