Reminder of the Definitions:

(1) For a and b real numbers with a < b, (a,b) ={ x : a < x <
b}

(2) A set of real numbers, O, is called an open set if and only if
for any number x that is a member of O there are some numbers a and b so
that x is a member of (a,b) and (a,b) is a subset of O.

**Proposition**: If U and V are open sets, then U intersect
V is an open set.

**Proof**: Suppose x is in U intersect V. Then x is in U and x is
in V.

Since x is in U and U is an open set, there are some numbers a and b where x is a member of (a,b) and (a,b) is a subset of U. Since x is in V and V is an open set, there are some numbers c and d where x is a member of (c,d) and (c,d) is a subset of V.

Let p denote the maximum of the numbers a and c and let q be the minimum of the numbers b and d, then x is a member of (p,q) and (p,q) is a subset of U intersect V.

Thus U intersect V is an open set. **EOP.**

1. Is the statement in the proposition conditional or absolute? If conditional, what are the hypotheses and conclusions? If absolute, can you rephrase the statement as a conditional statement?

2. List the variables used in the proposition. Indicate what these variables repesent. (numbers? sets?...)

3. Is the proof of this proposition direct or indirect?

a. If the proof is indirect, state the way in which the argument proceeds.(What is assumed? What is actually demonstrated?)

b. If the proof is direct, does the proof proceed forward or is it mixed with some backward argument? If it has some backward argument, indicate briefly how the original conclusion is altered.

4. Did the proof explicitly leave some steps for the reader to complete? If so, state what steps the reader is expected to complete. [Optional: complete these steps.]

5. Did the proof implicitly leave some steps for the reader to complete?
If so, state what steps you think the reader is expected to complete. [Optional:
complete these steps.]

6. Indicate any parts of the argument that you felt needed greater detail
or better connection to the proof. [Optional: Supply these details or suggest
a better connection.]

7.a. Prove {x: x > a} and {x : x < b} are open sets.

b. Using part a and the proposition, prove that if a<b
then (a,b) is an open set.

8. Overall, do you think this proof was effective? Discuss briefly the basis for you conclusion.