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qn 56
3. Every Cauchy sequence is convergent.
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4. Every non-empty set of real numbers which is
bounded above has a least upper bound.
Historical note
About 500 B.C. the Pythagoreans showed that the sides of a 45, 45,
90 triangle were incommensurable. This is equivalent, in modern terms,
to proving that
2 is irrational. A multiplicity of irrationals is discussed
in Euclid, Book X. The treatment of ratios given in Euclid, Book V (c.
300 B.C.) includes a careful description (in definition 5) of how to
compare two incommensurable lengths. Until the nineteenth century,
European mathematicians took their notions of number from a
geometric view of measurements.
Eighteenth century writers implicitly assumed that bounded
monotonic sequences were convergent in their discussion of series and
also assumed that lines which crossed on a graph necessarily had a
point of intersection.
Thedscription of a
least upper bound
which is not attained appears
in thethsis of C. F. Gauss (1799), and thenotion of lim sup and lim inf,
geometrically defined, in an unpublished notebook of his about 1800.
The term 'la plus grande des limites' is used by Cauchy (1821). The
notation, lim sup and lim inf, was introduced by Pasch in 1887.
In 1817 B. Bolzano proved that the convergence of Cauchy
sequences implied the least upper bound property, though his proof
that Cauchy sequences converged was defective since he had nothing
equivalent to an axiom of completeness.
In 1821, A. L. Cauchy proved that the partial sums of a convergent
series (see chapter 5) satisfy the Cauchy criterion and claimed that partial
sums satisfying the Cauchy criterion were those of a convergent series,
but without proof. Cauchy also presumed that bounded monotonic
sequences were convergent, both in his work on series and in his work
on continuity. Cauchy stated that irrational numbers were limits of
sequences of rational numbers, but did not argue from this proposition.
Until the1860s threwas no public discussion to clarify thestatus
of irrational numbers. In his lectures in Berlin in 1865, K. Weierstrass
gave an elaborate construction of positive irrational numbers as
bounded infinite sums of rational numbers. Weierstrass insisted on the
fundamental importance of the theorem that an infinite bounded set has
a cluster point, which he proved in his lectures from about 1867.
In attempting to simplify the treatment of Weierstrass, both E.
Heine and G. Cantor defined real numbers as the limits of rational
