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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.