Definitions and statements of theorems, propositions, lemmas, or corollaries need not be formulated in a specific way, but should be equivalent to the versions seen in the course. In theorems, propositions, lemmas, or corollaries, you must give both the hypothesis and the conclusion, and you must clearly distinguish between the two.
In proofs, you should above all convey the underlying ideas, but for full marks, correct reasoning is required. Take the lecture notes as an example for the level of detail expected; sometimes you don't need to give all the details, especially for routine calculations, but you should still mention these statements.
As this is a new unit, there are no past exam papers available from the library. However, you will find many questions in past exam papers for MA30041 (Metric Spaces) and MA40043 (Real & Abstract Analysis) that are suitable to test your level of preparation for this unit. A selection of questions mostly from these past exam papers are collected in a mock exam available here. Solutions are available here. (I do not recommend that you rely on past exam papers for studying the subject, as they are not designed for this purpose, but they may be useful to test yourself.)