I would say some author would like to use "there exist" as a direct translation of the existential quantifier. In this case, like in your case, the author does not bother to fit his writing in a plain English context. Note that if your readers are all mathematicians or people with pure mathematics background, then you need not worry about such a usage would confuse your reader.
Nevertheless, I prefer to write, for example: "For every real number ϵ there is a real number δ such that ..." or "There are integers x, y, z such that x2 + y2 = z2 " or "There is a triple of integers $x, y, z$ such that x2 + y2 = z2.
The case that interests you is not the only case. For instance, we can easily find that some author would write "Given ϵ > 0, ..." instead of "Given an ϵ > 0".
A beautiful mathematical writing should be like the author is talking and walking with the reader.
If you are interested, please be referred to How to write mathematics