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III - Convergence: Continuous variables
for every finite subset F of I. By hypothesis, the KF are, like the Ji , nonempty
compact intervals. If, further, F' and F" are two finite subsets of I and if
F = F' U F", then (associativity of intersections)
KF = n Ji = n Ji , n n Ji" = KF' n KFII,
iEF'UF"
i'EF'
i"EF"
so that the family of K F , indexed by the set of all finite subsets of I, satisfies
the hypothesis of Theorem 10. The intersection of all the K F , i.e. of all the
Ji , is thus a nonempty compact interval, qed.
If for example one has a countable family (In), the hypothesis means that
J1 n ... n I n = In is nonempty for any n, since every finite subset of N is
contained in a set {I, ... , n}. In this case, Corollary 2 follows directly from
the previous one.
Corollary 3 (Bolzano-Weierstrass Theorem). Every bounded sequence
of complex numbers has a convergent subsequence.
First we examine the case of a sequence u(n), n E N, with real terms,
so contained in a compact interval I of radius r. We divide I into two equal
compact intervals (they have a point in common, but this does not matter)
and, for each of them, we consider the set of n E N such that u( n) belongs to
it. We obtain two subsets of N whose union is N. One of these at least, say
N1, is an infinite set; let us denote by It that of the two halves of I containing
the u( n) corresponding to N 1. After this, let us divide again It into two equal
compact intervals; the preceding argument shows that, for one of these two
halves, say 12, the set N2 of n E Nl such that u(n) E 12 is infinite. This done,
let us again divide, 12 into two equal halves, etc.
In this way we define a decreasing sequence of compact nonempty intervals
h c I and a decreasing sequence of infinite subsets Nk of N satisfying the
following conditions:
(i) u(n) E Ik for all n E Nk ;
(ii) h has radius r /2k
where r is the radius of I. By Corollary 1 above, the h have a common point
u, clearly unique by (ii).
Write PI for the least element of N1 . Every element of N2 is ~ PI since
N2 c N1 ; since N2 is infinite, it contains numbers strictly greater than PI;
write P2 > PI for the least of these. Generally, write Pk for the least of the
numbers of Nk which is > Pk-l. Finally, write v(k) = U(Pk), a subsequence
of the given sequence.
Relation (i) above shows that v(k) E Ik for all k, and so satisfies
Iv(k) - ul < r/2k. Thus the subsequence v(k) converges to u, which completes the proof for real sequences.
III - Convergence: Continuous variables
for every finite subset F of I. By hypothesis, the KF are, like the Ji , nonempty
compact intervals. If, further, F' and F" are two finite subsets of I and if
F = F' U F", then (associativity of intersections)
KF = n Ji = n Ji , n n Ji" = KF' n KFII,
iEF'UF"
i'EF'
i"EF"
so that the family of K F , indexed by the set of all finite subsets of I, satisfies
the hypothesis of Theorem 10. The intersection of all the K F , i.e. of all the
Ji , is thus a nonempty compact interval, qed.
If for example one has a countable family (In), the hypothesis means that
J1 n ... n I n = In is nonempty for any n, since every finite subset of N is
contained in a set {I, ... , n}. In this case, Corollary 2 follows directly from
the previous one.
Corollary 3 (Bolzano-Weierstrass Theorem). Every bounded sequence
of complex numbers has a convergent subsequence.
First we examine the case of a sequence u(n), n E N, with real terms,
so contained in a compact interval I of radius r. We divide I into two equal
compact intervals (they have a point in common, but this does not matter)
and, for each of them, we consider the set of n E N such that u( n) belongs to
it. We obtain two subsets of N whose union is N. One of these at least, say
N1, is an infinite set; let us denote by It that of the two halves of I containing
the u( n) corresponding to N 1. After this, let us divide again It into two equal
compact intervals; the preceding argument shows that, for one of these two
halves, say 12, the set N2 of n E Nl such that u(n) E 12 is infinite. This done,
let us again divide, 12 into two equal halves, etc.
In this way we define a decreasing sequence of compact nonempty intervals
h c I and a decreasing sequence of infinite subsets Nk of N satisfying the
following conditions:
(i) u(n) E Ik for all n E Nk ;
(ii) h has radius r /2k
where r is the radius of I. By Corollary 1 above, the h have a common point
u, clearly unique by (ii).
Write PI for the least element of N1 . Every element of N2 is ~ PI since
N2 c N1 ; since N2 is infinite, it contains numbers strictly greater than PI;
write P2 > PI for the least of these. Generally, write Pk for the least of the
numbers of Nk which is > Pk-l. Finally, write v(k) = U(Pk), a subsequence
of the given sequence.
Relation (i) above shows that v(k) E Ik for all k, and so satisfies
Iv(k) - ul < r/2k. Thus the subsequence v(k) converges to u, which completes the proof for real sequences.