I've noticed that some graduate students never learned how to write a formal or at least a clear definition of a concept or a problem description. So instead of something like
For all $n \in \mathbb{N}$, let a function $f_n : \mathbb{R} \rightarrow \mathbb{R} $ be defined as $f_n(x) = x^n$. We prove that for all even values of $n$ and for all real-valued $x>0$, we have $f_n(x)>0$.
they would something write:
We need every function to have $f_n(x)>0$. For this, $x>0$. If indeed $x>0$, then for such a $x$ we have $f_n(x) = x^n$. This is what we can prove while $n$ is even.
(Note that this is made-up example, and I am not claiming that my "good" example is perfect -- it is just good enough for many purposes)-
Are there any introductory texts that can be given to Master thesis students to improve their formal writing and that probably contain a few exercises that they can do to get started?
I've found a few documents on mathematical writing, but they are targeted towards readers who already know how to structure their arguments in a clear way. I'm searching for something more on the basic level.
You might try a tutorial based on any automated proof checker. It will accept ONLY formal definitions.
In my experience, however, not all formal definitions are created equal. A definition may be formal but quite useless for the purposes at hand. And its usefulness will only become apparent when you try to use it in proofs. Subtle changes can make a big difference in this regard. To drive home this point, you should require the student, at the very least, to use their formal definition and the proof checker to actually derive some elementary, even trivial result based on their definition, e.g. define an infinite set and prove that the set of natural numbers is infinite by that definition. Or prove that the empty set is not infinite (i.e. it is finite).
May I humbly suggest my proof checker that is available free at my homepage.