Question: I recently marked over $100$ assignments for a multivariable calculus course. One question which a lot of people did poorly was proving a given set was open. Aside from issues relating to rigour and logic (or lack thereof), I noticed an issue that I wasn't as aware of before this experience. A lot of students used English words incorrectly (from a mathematical point of view) when writing sentences within their proof.

Some of the words/phrases I am referring to are:

* As
* Assume
* Let
* Suppose
* Hence
* Therefore
* Such that
* There exists

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> Is anyone aware of a reference which explains how to use these words and phrases (and others that frequently occur) in a mathematical context?
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Answer: 
Again ('cos I've recommended it before) I can very warmly recommend getting your students to beg/borrow/buy and then **read** the excellent

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> Daniel J. Velleman, *How to Prove it: A Structured Approach* (CUP, 1994 and much reprinted, and now into a second edition).
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From the blurb: "Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs." And in doing that, the most important thing, Velleman's book should get students to understand the lingo properly as they better understand what they are doing when they make a supposition here, draw an inference there, use a quantifier, etc.
I'm using

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> Franco Vivaldi *Mathematical Writing* (Springer Undergraduate Mathematics Series)
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which I find very useful. It contains many definitions of basic mathematical objects (sets, maps...) as well as examples of proves.