MthT 430 Notes Chapter 8g More Equivalent Statements of (P13)
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MthT 430 Notes Chapter 8g More Equivalent Statements of (P13)
Assuming (P1 - P12), there are several equivalent statements of the Least Upper Bound Property (P13).

(P13 or P13-LUB) - Least Upper Bound Property. If A is a non empty set of real numbers, and A is bounded above, then A has a least upper bound.

(P13-BIN) Binary Expansions Converge. Every binary expansion represents a real number x: every infinite series of the form
c1 2−1 + c2 2−2 + …,    ck ∈ {0,1},
converges to a real number x, 0 ≤ x ≤ 1.

(P13-DECIMALS)- Decimal Expansions Converge. Every decimal expansion represents a real number x: every infinite series of the form
c1 10−1 + c2 10−2 + …,    ck ∈ {0,…,9},
converges to a real number x, 0 ≤ x ≤ 1.
The equivalence of (P13) and (P13-BIN) is shown in chap8a.tex.
See homepages.math.uic.edu/~jlewis/mtht430/chap8a.htm#BIN

(P13-BISHL) - Bounded Increasing Sequences Have Limits. Let {xn}n=1∞ be a bounded monotone increasing sequence; i.e.
x1 ≤ x2 ≤ …,
and there is a number M such that for n = 1,2, …,
xn ≤ M.
Then there is a number L such that

lim
n → ∞ 
 xn = L.
 Note that

lim
n → ∞ 
 xn
=
sup
n 
 xn.
(P13-BIN) implies (P13-BISHL) is shown in chap7c.tex.
See homepages.math.uic.edu/~jlewis/mtht430/chap7c.htm#BISHL

(P13-CFCBIB) - Continuous Functions on Closed Bounded Intervals are Bounded. If f is continuous on [a,b], then f is bounded above on [a,b], that is, there is some number N such that f(x) ≤ N for all x in [a,b].
CFCBIB2BISHL: (P13-CFCBIB) implies (P13-BISHL) is shown as follows (Thanks to a hint from Brayton Gray):
Proof. The proof is by contradiction. Suppose (P13-BISHL) is false. Then there is a bounded strictly increasing sequence, {xn}, which does not have a limit or sup. Let 0 = x0 < x1 < x2 < …, be bounded above by, say, [1/2]. Let
B
={x ∈ [0,1] | x is an upper bound for {xn}},
A
= [0,1] \B.
Then B and A are both nonempty and open in [0,1].
For x ∈ [0,1], define
f(x)
= 0, x ∈ B,
f(xk)
= 2k, k = 0, 1, …,
f(x)
= f(xk) + f(xk+1) − f(xk)
xk+1 − xk
 (x − xk), linear, xk ≤ x ≤ xk+1.
Then f is a continuous function on [0,1] which is unbounded and (P13-CFCBIB) is not satisfied.



(P13-BW) - Bolzano-Weierstraß Property. Let {xn}n=1∞ be a sequence of points in. [0,1]. Then there is an x in [0,1] which is a limit point1 of the sequence {xn}n=1∞.
(P13-BIN) implies (P13-BW) was shown in chap7b.tex.
See homepages.math.uic.edu/~jlewis/mtht430/chap7b.htm#BW
Other Statements Equivalent to (P13)
Assuming (P1 - P12), there are other statements equivalent to (P13):

(P13-CFIVP) - Continuous Functions on Intervals Have the Intermediate Value Property. If f is continuous on [a,b] and f(a) < 0 < f(b), then there is some x in [a,b] such that f(x) = 0.
(P13-BIN) implies (P13-CFIVP) was shown in chap7b.tex.
See homepages.math.uic.edu/~jlewis/mtht430/chap7b.htm#CFIVP

(P13-CFCBIMAX) - Continuous Functions on Closed Intervals assume a Maximum Value for the Interval. If f is continuous on [a,b], then there is a number y in [a,b] such that f(y) ≥ f(x) for all x in [a,b].

(P13-HB) - Heine-Borel Theorem. Every open cover of a closed bounded interval contains a finite subcover of the closed interval.

(P13-CAUCHY) - Cauchy Sequences Have Limits. If {xn} is a Cauchy sequence2, then there is a number x such that

lim
n→∞ 
 xn = x.
This property is often stated: The real numbers are complete .

(P13-CFCIUC) Continuous Functions on Closed Bounded Intervals are Uniformly Continuous. If f is continuous on [a,b], then f is uniformly continuous on [a,b]. See Spivak, p. 143.

(P13-CONNECTED) - Intervals are Connected. An open [closed] interval cannot be decomposed into two disjoint nonempty open [closed] subsets.
With a little bit of topology, (P13-CFIVP) can be shown to be equivalent to (P13-CONNECTED).

Footnotes:

1A point x is a limit point of the sequence if for every ϵ > 0, infinitely many terms of the sequence are within ϵ of x. Alternately, there is a subsequence which converges to x. A more informal idea is to say that infinitely many terms are as close as desired to x.
2Look up the definition of Cauchy sequence. A working def inition given by Konrad Knopp in Introduction to the Theory of Functions is that almost all the terms are close together .

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