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What is a mathematical proof?
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A proof is a sequence of logical statements, one implying another, which gives an explanation of why a given statement is true. Previously established theorems may be used to deduce the new ones; one may also refer to axioms, which are the starting points, "rules" accepted by everyone. Mathematical proof is absolute, which means that once a theorem is proved, it is proved for ever. Until proven though, the statement is never accepted as a true one.

Writing proofs is the essence of mathematics studies. You will notice very quickly that from day one at university, lecturers will be very thorough with their explanations. Every word will be defined, notations clearly presented and each theorem proved.

We learn how to construct logical arguments and what a good proof looks like. It is not easy though and requires practice, therefore it is always tempting for students to learn theorems and apply them, leaving proofs behind.

This is a really bad habit (and does not pay off during final examinations!); instead, go through the proofs given in lectures and textbooks, understand them and ask for help whenever you are stuck. There are a number of methods which can be used to prove statements, some of which will be presented in the next sections.

Hard and tiring at the beginning, constructing proofs gives a lot of satisfaction when the end is reached successfully.

"The search for a mathematical proof is the search for a knowledge which is more absolute than the knowledge accumulated by any other discipline." - Simon Singh

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