Writing a proof consists of several phases. Write down all of the steps carefully, without skipping any of them. Some of the first stages are frequently (but not always) the supplied assertions, and the last step is the conclusion that you set out to establish.

The always required first step in writing a proof is to state **your assumptions**. Without assumptions, there is no way to begin to prove or disprove anything. Therefore, they should be stated clearly at the beginning of the proof. They can be as simple as "let $a$, $b$ be integers such that $a < b$", or as complex as "for all positive integers $n$, let $f(n)$ be the highest power of 2 less than or equal to $n$. Show that $\lim_{n \to \incomma 0} f(n)/n = 1$". Always start with **clear and simple statements** that can be easily proved or refuted.

After making sure that your assumptions are correct, you can start proving facts about them that will help you build up to the conclusion. For example, if one of your assumptions is that "$a$, $b$ are integers such that $a < b$", then you could prove **some properties** about these numbers that have implications for whether or not they satisfy this assumption.

Writing a proof consists of several phases.

- Draw the figure that illustrates what is to be proved.
- List the given statements, and then list the conclusion to be proved.
- Mark the figure according to what you can deduce about it from the information given.

The Proof's Structure

- Draw the figure that illustrates what is to be proved.
- List the given statements, and then list the conclusion to be proved.
- Mark the figure according to what you can deduce about it from the information given.
- Write the steps down carefully, without skipping even the simplest one.

It is sometimes simpler to jot down the assertions first, then go back and fill in the justifications afterwards. At times, you will simply write assertions and reasons at **the same time**. There is no one-size-fits-all procedure for evidence, just as there is no one-size-fits-all length or order for claims. The goal is to use whatever proof method best fits the nature of the problem.

In mathematics, a proof is a sequence of arguments intended to demonstrate that a given statement called the conclusion is indeed true. A mathematical argument is valid if neither its converse nor any of its subarguments can be shown to be false. For example, if it were possible to prove that every integer greater than 2 must be divisible by at least one of 2, 3, or 4, then we could conclude that every integer is divisible by at least one of 2, 3, or 4 and thus that all integers can be written as products of pairs of numbers of which at least one is odd, one even. This would be a valid proof because it does not contain any statements or assumptions that are themselves not true.

Mathematics is unique among the sciences in that almost anything can be proven using **only formal logic** and mathematical reasoning. No matter how counterintuitive this may seem, it is a fundamental tenet of the discipline that nothing can be proved within the system unless it already has been assumed or taken for granted.

Choose many statements with **simple proofs** from the textbook to learn how to do proofs. Make a note of the claims but not the evidence. Then try if you can back it up. Students sometimes attempt to prove a proposition without employing the complete premise. For example, they might say "Let x be any number". They cannot say anything meaningful; therefore, the claim is meaningless. You should avoid this pitfall.

To prove a statement, first you need to know that it is true. Second, you need to come up with a proof. There are two ways to do this: reductio ad absurdum and argument from authority. With reductio ad absurdum, you assume that the statement is false and use one of its parts to conclude that something extremely bad would happen if it were true. For example, let's say you want to prove that all men are mortal. You could say that some immortal man has been born, which would be a contradiction so all men are mortal. This method can only be used with statements that are easy to refute this way. For example, if you were to use reductio ad absurdum to prove that all dogs are mortal, you would need a very large group of animals for someone to have escaped mortality by being an exception.

The second way to prove a statement is through argument from authority.

Each proof consists of seemingly accurate stages, yet each argument contains a flaw that leads to an illogical outcome. These proofs may persuade you, and your common sense may be attacked. A good critical thinker will question the logic behind **these proofs** and look for **alternative explanations** that could have been used but weren't. One should always keep in mind that there are bad arguments and poor reasoning skills, so it's important to learn how to identify them.

Bogus proofs often do one or more of the following: they use invalid forms of logic, they make assumptions without checking whether those assumptions are true, they mix up different kinds of arguments, they ignore relevant facts or evidence, they claim too much, they rely on personal opinions or prejudices rather than proven facts, they use language that has no clear meaning, etc.

In conclusion, bogus proofs can be very convincing, but they're only that good. Always remember that there are many ways to prove something, and not all proofs are valid. With practice, you will be able to distinguish between **real and bogus proofs**.

Write the first paragraph extremely carefully. Write out the definitions very precisely, as well as the things you are authorized to assume, in meticulous mathematical terminology. Write the conclusion extremely carefully. That is, in meticulous mathematical terminology, put down what you are attempting to establish. Then go back and fill in any gaps of logic or certainty.

Writing a good mathematical proof is much like writing a persuasive essay. You need to be able to explain your ideas clearly and accurately, and not rely on intuition or subjective judgement. Mathematical proofs are different from other kinds of essays in many ways, but the key thing is that they must make sense: if you write something that makes no sense, then nobody will be able to understand it. So always try to keep this question in mind when writing or thinking about proofs: what would happen if this were not the case?

There are several different types of proofs, but they can generally be divided into **four main categories**: logical proofs, geometric proofs, algebraic proofs, and numerical proofs. Logical proofs start with **certain assumptions** and then use **these assumptions** to conclude that some statement is true. For example, suppose we want to prove that all men are mortal. We could start by assuming that there is no immortal man (a false assumption) and then conclude that all men are mortal by applying the concept of immortality to men in general.