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
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
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.
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.
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
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
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