I have to thank one of my undergraduate math professors for teaching me this invaluable lesson when I first began learning analysis. These are some words which are ubiquitous throughout mathematics and physics even Nobel laureates and top journals use them, and there is no excuse for using them other than laziness and bad style. Math "bad words": "Let" - ambiguous. It can mean "suppose", "choose", or "define" depending on the situation. "Where" - lazy style. Always define your variables before using them. "If...then..." and implication arrows: almost never used correctly. The only time it is safe to use these is in a heading to a theorem/lemma. Use the following to make your proofs much more clear, precise, and readable: Need to make a "for every ..." statement? "Suppose..." Know something exists and want to use it? "Choose..." Need to make a new definition? "Define..." Miscellaneous helping words: "such that": a qualifier, enables more precision. "Since": helps make a justification for a passive statement "using the fact that": like "since," but when you are using a piece of information to actually DO something. The above 6 English words/phrases are just about all you need to write any proof in mathematics (besides mathematical symbolism, of course). Any other words added are just for making a proof look prettier.