- Let S = . Find supS. Now let S = [2,5). Find supS. Is supS always a number in S? Give conditions under which supS ∈ S and then give conditions under which inf S ∈ S.
- Show that if S≠∅ and is bounded above (below) then supS (inf S) is unique. That is, there is only one least upper bound and only one greatest lower bound. If S = ∅ can you conclude that 7 is an upper bound? Can you conclude 7 is a lower bound? What about 13.5? What about any other number?
- Let S be a set which is bounded above and let −S denote the set . How are infand suprelated? Hint: Draw some pictures on a number line. What about supand inf S where S is a set which is bounded below?
- Which of the field axioms is being abused in the following argument that 0 = 2? Let
x = y = 1. Then
and so

Now divide both sides by x − y to obtain

- Give conditions under which equality holds in the triangle inequality.
- Let k ≤ n where k and n are natural numbers. P, permutations of n things taken k at a time, is defined to be the number of different ways to form an ordered list of k of the numbers,. Show
- Using the preceding problem, show the number of ways of selecting a set of k things
from a set of n things is .
- Prove the binomial theorem from Problem 7. Hint: When you take
^{n}, note that the result will be a sum of terms of the form, a_{k}x^{n−k}y^{k}and you need to determine what a_{k}should be. Imagine writing^{n}=where there are n factors in the product. Now consider what happens when you multiply. Each factor contributes either an x or a y to a typical term. - Prove by induction that n < 2
^{n}for all natural numbers, n ≥ 1. - Prove by the binomial theorem and Problem 7 that the number of subsets of a given
finite set containing n elements is 2
^{n}. - Let n be a natural number and let k
_{1}+ k_{2}+k_{r}= n where k_{i}is a non negative integer. The symboldenotes the number of ways of selecting r subsets of

which contain k_{1},k_{2}k_{r}elements in them. Find a formula for this number. - Is it ever the case that
^{n}= a^{n}+ b^{n}for a and b positive real numbers? - Is it ever the case that = a + b for a and b positive real numbers?
- Is it ever the case that =+for x and y positive real numbers?
- Derive a formula for the multinomial expansion,
^{n}which is analogous to the binomial expansion. Hint: See Problem 8. - Suppose a > 0 and that x is a real number which satisfies the quadratic equation,
Find a formula for x in terms of a and b and square roots of expressions involving these numbers. Hint: First divide by a to get

Then add and subtract the quantity b

^{2}∕4a^{2}. Verify thatNow solve the result for x. The process by which this was accomplished in adding in the term b

^{2}∕4a^{2}is referred to as completing the square. You should obtain the quadratic formula^{2},The expression b

^{2}− 4ac is called the discriminant. When it is positive there are two different real roots. When it is zero, there is exactly one real root and when it equals a negative number there are no real roots. - Suppose f= 3 x
^{2}+ 7x − 17. Find the value of x at which fis smallest by completing the square. Also determine fand sketch the graph of f. Hint: - Suppose f= −5x
^{2}+ 8x − 7. Find f. In particular, find the largest value of fand the value of x at which it occurs. Can you conjecture and prove a result about y = ax^{2}+ bx + c in terms of the sign of a based on these last two problems? - Show that if it is assumed ℝ is complete, then the Archimedean property can be proved.
Hint: Suppose completeness and let a > 0. If there exists x ∈ ℝ such that
na ≤ x for all n ∈ ℕ, then x∕a is an upper bound for ℕ. Let l be the least
upper bound and argue there exists n ∈ ℕ ∩. Now what about n + 1?
- Suppose you have an infinite sequence of numbers a
_{1}≤ a_{2}≤. Also suppose there exists an upper bound L such that each a_{k}≤ L. Review what it means for the limit of a sequence to exist and verify that lim_{n→∞}a_{n}= sup. In other words, the limit equals the least upper bound of the numbers.

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