3. Inequalities and ordered ﬁelds 13

bound (inﬁmum). We prove the ﬁrst, leaving the second to the reader. Sup-

pose for all n we have

x1 ≤ ... ≤ xn ≤ xn+1 ≤ ... ≤ M.

Let α be the least upper bound of the set {xn}. Then, given 0, the number

α - is not an upper bound, and hence there is some xN with α - xN ≤ α. By

the nondecreasing property, if n ≥ N, then

(9) α - xN ≤ xn ≤ α α + .

But (9) yields |xn-α| and hence provides us with the needed N in the deﬁnition

of the limit. Thus limn→∞(xn) = α.

x2 x1 x3 x4 M

α

Figure 1.4. Monotone convergence.

Remark 3.2. Let {xn} be a monotone sequence of real numbers. Then {xn}

converges if and only if it is bounded. Proposition 3.3 guarantees that it converges

if it is bounded. Since a convergent sequence must be bounded, the converse holds

as well. Monotonicity is required; for example, the sequence

(-1)n

is bounded but

it does not converge.

The next few pages provide the basic real analysis needed as background ma-

terial. In particular the material on square roots is vital to the development.

I

Exercise 1.16. Finish the proof of Proposition 3.3; in other words, show that

a nonincreasing bounded sequence converges to its greatest lower bound.

I

Exercise 1.17. If c is a constant and {xn} converges, prove that {cxn} con-

verges. Try to arrange your proof such that the special case c = 0 need not be

considered separately. Prove that the sum and product of convergent sequences are

convergent.

I

Exercise 1.18. Assume {xn} converges to 0 and that {yn} is bounded. Prove

that their product converges to 0.

An extension of the notion of limit of sequence is often useful in real analysis.

We pause to introduce the idea and refer to [20] for applications and considerably

more discussion. When S is a bounded and nonempty subset of R, we write as usual

inf(S) for the greatest lower bound of S and sup(S) for the least upper bound of

S. Let now {xn} be a bounded sequence of real numbers. For each k, consider

the set Xk = {xn : n ≥ k}. Then these sets are bounded as well. Furthermore

the bounded sequence of real numbers deﬁned by inf(Xk) is nondecreasing and the

bounded sequence of real numbers sup(Xk) is nonincreasing. By the monotone

convergence theorem these sequences necessarily have limits, called lim inf(xn) and

lim sup(xn). These limits are equal if and only if lim(xn) exists, in which case