"Proof is an idol before whom the pure mathematician tortures himself."
 Arthur Stanley Eddington, The Nature of the Physical World
The fact that a theorem (in mathematics) has been proved allows us to rely on its results without fear of exceptions. But what does it mean for a theorem to be proved? What is a proof?
DEFINITION  Proof
A proof, in mathematics, is the validation of a proposition or theorem by application of specified rules in a series of logical steps.
A proof, in mathematics, is the validation of a proposition or theorem by application of specified rules in a series of logical steps.
This basically means that a proof is a very carefully constructed set of arguments laid out so that nobody, no matter how smart or creative, could refute the logic.
Let’s show how a proof is done by using a series of logical arguments to prove the following theorem.
Let’s show how a proof is done by using a series of logical arguments to prove the following theorem.
THEOREM
If it is raining, then it must be cloudy.
PROOF
Let us first assume that it is raining. Then, by the definition of rain, we know that we have water falling from the sky. However, if water is falling from the sky, then there has to be moisture in the sky (since water is moisture). For there to be enough moisture in the sky to cause this moisture to coalesce into rain drops, there must be clouds. Therefore, if it is raining, it must be cloudy.
If it is raining, then it must be cloudy.
PROOF
Let us first assume that it is raining. Then, by the definition of rain, we know that we have water falling from the sky. However, if water is falling from the sky, then there has to be moisture in the sky (since water is moisture). For there to be enough moisture in the sky to cause this moisture to coalesce into rain drops, there must be clouds. Therefore, if it is raining, it must be cloudy.
Okay, I am not a meteorologist so I may have butchered that last bit of the proof; however, if I did start using more technical terms from meteorological science, I may bore you to death. I sincerely hope that example delivers the idea of a proof.
A subtle note about proofs is required here. The proof is necessary for a conjecture to be classified as a theorem; however, the proof is not considered part of the theorem. This basically means that proofs and theorems are different beasts. When someone in mathematics states a theorem, you have the right to request a proof of their statement; however, it is not necessary.
Every effort is made to prove all theorems stated in this textbook and you should genuinely try to understand how each proof works; however, there may be times when a proof will be beyond the scope of our skills. For these theorems I will mention that more advanced work is needed to derive the proof or I may include their proofs in the appendix. Let’s try another proof to get the hang of it. 
Concept 
THEOREM
If you owned the car called the Aurora, then you would own the only Aurora in the world. 

PROOF
Suppose that you owned the car called the Aurora. Since the designer only built one of these cars, then you would own the only Aurora in the world. 
Historical Note 
We might as well go for one more while we are at it. Prove the following theorem.
THEOREM
Every president with the last name of Adams finished his presidency in the 19th century.
Every president with the last name of Adams finished his presidency in the 19th century.
In this case, we first rewrite the theorem as an "if…, then…" statement so that we are really trying to prove the equivalent statement,
"If a president has the last name of Adams, then he finished his presidency in the 19th century."
PROOF
The presidents with the last name of Adams are John Adams and John Quincy Adams. John Adams finished his presidency in 1801 and John Quincy Adams finished his presidency in 1829. Therefore, all presidents with the last name of Adams finished their presidency in the 19th century.
The presidents with the last name of Adams are John Adams and John Quincy Adams. John Adams finished his presidency in 1801 and John Quincy Adams finished his presidency in 1829. Therefore, all presidents with the last name of Adams finished their presidency in the 19th century.
Before walking away from our discussion of theorems, I would like to mention that there are two subtypes of theorems we encounter in mathematics: the lemma and the corollary. At this point, the distinction between a theorem, a corollary, and a lemma is somewhat arbitrary. We will define these when we encounter them later in the textbook; however, you could start a new skill by looking in the index for the word lemma and reading its definition right now.

