Assuming (P1 - P12), there are several equivalent statements of
the Least Upper Bound Property.
(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 + ¼, ck Î {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 + ¼, ck Î {0,¼,9},
converges to a real number x, 0 £ x £ 1.
The equivalence of (P13) and (P13-BIN) is shown in chap8a.tex.
See
http://www.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 £ ¼,
and there is a number M such that for n = 1,2, ¼,
xn £ M.
Then there is a number L such that
lim
n ® ¥
xn = L.
(P13-BIN) implies (P13-BISHL) is shown in chap7c.tex.
See
http://www.math.uic.edu/~jlewis/mtht430/chap7c.htm#BISHL (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
http://www.math.uic.edu/~jlewis/mtht430/chap7b.htm#BWOther Statements Equivalent to (P13-LUB)
Assuming (P1 - P12), there are other statements equivalent to
(P13-LUB):
(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
http://www.math.uic.edu/~jlewis/mtht430/chap7b.htm#CFIVP (P13-CFCIB) - Continuous Functions on Closed
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].
(P13-CFCIMAX) - 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) ³ f(x) for all x in [a,b].
(P13-HeineBorel) - Heine-Borel Theorem. Every open cover of a closed 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) Closed Intervals are Connected3.
Footnotes:
1A point x is
a limit point of the sequence if for every e > 0, infinitely
many terms of the sequence are within e 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 definition given by
Konrad Knopp in Introduction to the Theory of Functions is
that almost all the terms are close together .3Look up the
definition of a connected set.
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