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: 
Charles Wells, *The Handbook of Mathematical Discourse*; it’s available as a PDF [here](https://www.abstractmath.org/Handbook/handbook.pdf). His site [abstractmath.org](http://www.abstractmath.org/MM/MMIntro.htm) may also be useful.
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.