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Rules of Proof

Every statement must be justified.
A justification can refer to prior lines of the
proof, the hypothesis and/or previously proven statements
from the book.
Cases are often required to complete a proof which has
statements with an "or" in them. Each case should be labelled
I, II etc and then the line numbering for each case should
start with the same number. Remember, you may justify any
statement with an earlier line in the proof (but those earlier lines
cannot be from other cases).
You may never assume anything except when doing a
proof by contradiction. When writing a proof by contradiction
the first line is "Assume on the contrary" and then write
the negation of the conclusion of what you are trying to prove.
A contradiction is reached when a statement contradicts any of the
hypotheses, a prior line of the proof, or any known fact (e.g. 1>0).
A proof by contradiction is only complete when a contradiction
has been reached in every case.
If you cannot find a justification and you are sure
it is true, you may wish to write a seperate proof on the
side to prove that justification and then apply it to your
original situation. When you do this, it is called proving
a lemma.
To prove a statement of the form, "For all x in some set
there exists y in some other set such that...", think of
x as given information and the fact that it is in a certain set
as part of the hypothesis, then in your proof proceed to look
for a y. You are not given y, you need to find it and it
might be that you can think of a formula for y and then you must prove
that this y has all the properties described after the "such that".
During a proof you may introduce new variables by saying "Let x be...".
You can do this for example by choosing x from a set (this is called the
Axiom of Choice) or by giveing a formula for the new variable x. You
may not introduce a variable that is given in the hypothesis or already
used earlier in the proof.
Be careful when writing a proof of a theorem you know well from
precalc or calc that you do not use the theorem itself in the
middle of the proof. That is called a circular proof and it is incorrect.
Similarly when proving something is a limit or an inverse, feel
free to use techniques from prior courses to figure out the idea
or the "answer" but then you still must write a proof of what you
are saying being careful to justify each step from statements in
the book.
When writing a proof by induction, always remember to write
"We will prove this by induction". Then write "when n=1"...
and check that case. Then write: "Given: statement with n"
and "Show: statement with (n+1) inserted where n was" and proceed to
prove this with a two column proof. The usual approach is
to try to rewrite one side of an equality in a format which looks like
the given statement and then apply the given statement to simplify
the equation. When you apply the given statement, you write "by the
induction hypothesis" to justify this step.
When doing a proof by induction be very careful to make sure that
everythign you say works for an arbitrary natural number. It cannot
just work for all n bigger than 2. It must include n=1. See
the "all horses are the same color" example on
Induction notes.