For a definition:

Part 1. Description of objects involved.

Part 2. Description of the property under consideration.

Part 3. The name of objects satisfying this property.

For example, the definition of derivative:

Part 1. Let f be a function and a be and interiour point of its domain.

Part 2. Assume that lim... exists.

Part 3. Then we say that f is diff. at a and f'(a)=lim...

Please note that if I ask to write the definition of a derivative and you write just formula f'(a)=lim... then I don't consider it as the solution. I immediately ask what is f, what is a, why limit exists. Of course, you will get partials marks, but it will be far from full mark.

For a theorem:

Part 1. Description of objects involved.

Part 2. Assumptions on the objects (hypothesis of a theorem).

Part 3. Conclusion.

For example, Rolle's Theorem:

Part 1. Let f be a function and a

Part 3. Then there is c in (a, b) such that f'(c)=0.

About solutions: please note that you have to explain (possibly briefly) each step. Don't afraid to write words! In particular if I see few lines of formulas I don't know what you mean. And I don't want to guess! For example, often I see something like this:

line 1: x > y > 0

then (without explanations) line 2: x^2 > y^2

What do you mean? How should I guess?

Is it "line 1 implies line 2" (which is correct)

Is it "line 1 is equivalent to line 2" (which is incorrect)

Is it "line 1 is implied by line 2" (which is also incorrect)

It is also very good to review statements involving "there exists", "for every", and, more important both of them. Also you should be comfortable with notions "implies" and "equivalent". Please look at Problem 2 in Drill Problems 2, Problem 3 in H/A 1, Problem 2 in H/A 4, Problems 1 and 3 in H/A 6, Problem 1 in Drill Problems 4 in MATH 117.