Please show some personal pride and professionalism with your hand-in work. It does not have to be typed, but it should be neat and legible (no smudges, cross-outs, or coffee stains). Do not hand in your scratch work. All proofs must be formal paragraph proofs as found in textbooks and published papers. The proofs the professor writes on the board often contain informalities that are unacceptable in a formal proof.
You should not use logical quantifier symbols unless you are writing a proof in mathematical logic. Other previously defined symbols pertinent to the branch of mathematics (algebra or analysis for example) and all symbols from set theory are allowable and desirable. You may not invent your own symbols or modify existing symbols! In mathematics precise, concise expression is valued over wordiness.
Use displayed expressions, especially if the expression is the key idea of a proof. Experience will help you judge what to display and what to write in-line. If you later need to refer to a displayed expression, give the expression a number and use the number as the reference. For example, "It follows from (1) that". Watch how your textbook author does it.
Before presenting a proof, always provide a statement of the proposition or theorem you are proving. Do not just write "page 156 #2. Proof...".
You should not state definitions or theorems within a proof. For example, do not say, "Since a and b are relatively prime (which means a and b have no common divisors other than 1)". Instead, just say, "Since a and b are relatively prime". You may assume the reader is mathematically competent. This means, in particular, the reader knows all definitions and knows the statements of theorems. So to reference a named theorem (e.g. Cauchy integral theorem) within a proof, just use its name ("By the Cauchy integral theorem"). Before using such a theorem, be sure your proof has already established that all the hypotheses of the referenced theorem are met.