A.3 Strat egies of Proving Theorems 607We shall find it co nvenient to refer to this schematic in horizontal form:Table A.12To prove a theorem is to demonstrate that its hypotheses do, in fact,
provide sufficient evidence to guarantee the truth of the conclusion. In addition
to the hypotheses, a proof m ay use any agreed-upon rules of logic, definitions of
the terms involved, theorems previously proved, and perhaps even results from
other agreed-upon areas of mathematics. In one sense, proving a theorem is
like presenting an argument in a court of law. In the case of proving a theorem,
however , we must show that our evidence is totally convincing- not just b eyond
reasonable doubt, but beyond all doubt. There is something absolute about a
proof in mathematics. In a mathematical proof we must show that if all our
assumptions are true, then our conclusion must be true (beyond any doubt).
No wonder we are studying a little logic as part of this course in real analysis!
Proving a theorem is an awesome responsibility.
Second, a theorem does not exist in isolation, but as part of a larger,
deductive system. It is derived from results established previously within the
system, uses terms defined within the system, and in turn is used in deriving
later results of the system. Thus, a theorem is like a node in a network of
results in which information flows in one direction only. In proving a theorem,
only information preceding it in the flow may be used. Beginning students often
have difficulty understanding that information to come later in the flow is never
allowed as a step, or a justification for a step, in the proof of a theorem.
We now present some proof strategies. We shall not give examples or ex-
ercises in this section. Indeed, this entire book serves as a set of examples of
these strategies. We set them forth here as reference for your use as you need
them.
PROOF STRATEGIES: PS-1-PS-7
(PS-1) DIRECT PROOF of Theorem: H 1 , H2, · · ·, Hn, .-.C.
"Direct proof" does not indicate a specific strategy as much as it indicates
that we are not using any of the specialized strategies to be described below.
We first gather together all that we know about the consequences of all the
hypotheses being true. We think about conditions that would make the conclu-
sion true. We try to link t he former to the latter. We describe this below as a
formal procedure. First, we need a theorem.
Theorem A.3.1 (Transitivity of Implication) If P => Q and Q => R, then
P=>R.