I have compiled a list of useful proof techniques for you, my fellow first-years, to assist you in proving results on the challenging problem sets we are confronting this year. If you have any comments or suggestions, please email me. Enjoy!

      
 
       METHODS OF CONFUSION:

       PROOF BY OBFUSCATION
               If you write with the intent of confusing the reader, you
       may succeed.  Since the grader cannot pinpoint where your
       answer deviates from the correct answer, she cannot mark your answer
       "wrong."

       PROOF BY HANDWAVING
               Very similar to obfuscation, but this approach at least
       attempts to SOUND like a proof.

       PROOF BY JARGON
               If you write as if you know what you are talking about, you
       may convince the reader that you are more knowledgeable than they
       are.  They will then be afraid to disagree or mark your answer "wrong."

       PROOF BY ILLEGIBILITY
               If the grader cannot read your answer, how can it be wrong?

       PROOF BY SATURATION
               If you write a many-page "proof", no one will bother to check.

       PROOF BY CONDENSATION
               The opposite of saturation.  You write an extremely terse
       proof in which you skip many steps (thereby avoiding the necessity
       of proving them!).

                       METHODS OF WISHFUL THINKING

       PROOF BY INTUITION
               If it feels good, go with it.

       PROOF BY REPUTATION
               Any idea you have ever had has already been thought of by
       Ken Arrow.  Since Arrow is always right, your idea must be true too.
       [I must acknowledge Ron B. as the source of this brilliant piece of
       logic!

       PROOF BY PRAYER
               If you pray that something is true, God might answer your
       prayer.

       PROOF BY INEBRIATION (OR HALLUCINATION)
               Everything always looks better when you're drunk.

       PROOF BY SELF-DEPRECATION
               Think to yourself: "I've proven this to be false, but I am
       ALWAYS wrong, so it MUST be true!"

       PROOF BY ANALOGY
               "A quasi-concave function is LIKE a concave function, so..."

       PROOF BY EXASPERATION
               If you stare at a partial, incorrect proof long enough, it
       slowly looks more and more like a complete, correct proof.

       PROOF BY ASSUMPTION
               If your assumptions are invalid, then your results are invalid
       too, so you might as well assume what you want to prove.  This saves
       time and frustration.

       PROOF BY CONSENSUS
               If everyone in the study group agrees, then it's got to be
       true!

       PROOF BY INTIMIDATION
               Threaten the listener/reader/grader with serious consequences
       if she disagrees (or speak in a threatening manner).  [E.g., Raphael
       vs. Murat... "This is BOGUS!"]

       PROOF BY TRIVIALIZATION
               How many times have you read in a textbook, "It is obvious
       that..." or "It can easily be shown that..." or, "The attentive reader
       will understand that..."  It is no coincidence that these statements
       are always made about the most difficult concepts in the book.  The
       author) could not prove them, so they trivialized them.  It's a very
       effective technique with widespread application and makes you look
       very, very smart in the process!

       PROOF BY SUPPOSITION
               Suppose that the statement you are trying to prove is true.
       Try to show a contradiction.  If you cannot, then the statement must
       be true!

                       METHODS OF CHEATING

       PROOF BY UPPER-YEAR STUDENTS
               Consult your favorite upper-year student who has been through
       the course already.

       PROOF BY LAST YEAR'S ANSWER SET
               Superior to consulting upper-year students, but useful only
       if they have bothered to save their notes.  The more confusing the
       class, the more likely they are to have burned their notes.

       PROOF BY PROOF
               Just find the result in a book somewhere and copy it.

                       METHODS OF TRANSFORMATION

       If you get a result that you don't like, there are a few special
       techniques which you might use to transform your intermediate result
       into more desirable final result.  These methods are applicable only
       in special cases, but when they work, they can be a powerful method of
       proof:

       PROOF BY INTERPOLATION:
               If you do two proofs and get two different answers and the
       answer that you want lies in between, just take the appropriate
       convex combination.

       PROOF BY PARENTHESES:
               Let's say that you have reduced the proof to solving the
       equation
                3 + 14 + 1 = 20.
       It may seem that you are stuck, but if you are observant, you will
       notice that adding parentheses will solve the problem:
                (3 + 1)(4+1) = 20

       PROOF BY TRANSFORMATION:
               This encompasses a few cases.  If you get X=2 and you
       need X=10, just switch to base 2.  Or, if you get X=0 and you
       need X=1, just exponentiate.  [These may seem to have limited
       application, but since many proofs yield X=0 or X=1, the ability
       to switch between them by taking logs or antilogs is a very useful
       technique!]

       PROOF BY SIGN CHANGE
               Just multiply one side of the equation by -1.

       PROOF BY ELIMINATION
               Just cross out any troublesome terms.

                       METHODS OF DELAY

       PROOF BY MATURATION
               Just tell the grader that you will prove this later when you
       are older and wiser.

       PROOF BY FERMAT
               "I had an elegant little proof of this, but this paper is not
       large enough to write it down..."

       PROOF BY TIME CONSTRAINT
               When you absolutely must turn in the problem set, your pseudo
       answer suddenly becomes a full proof.  You just scribble it down and
       you're done.  [Corollary: wait till the last minute!  You save lots
       of time this way.]