§3. Bolzano-Weierstrass and Cauchy's criterion
225
§3. Bolzano-Weierstrass and Cauchy's criterion
9 - Nested intervals, Bolzano-Weierstrass, compact sets
The purpose of this nO is to prepare the proof of Cauchy's general criterion
of convergence that we will establish in the next n°. But these preliminary
results are, even so, also important.
It all rests on the following result:
Theorem 10. Let (Kp)PEI an arbitrary18 family of compact nonempty intervals. Suppose that, for all p, q E I, there exists an rEI such that
Kr C Kp n K q. Then the intersection of the Kp is a compact nonempty
interval.
Let us write Kp = lap, bp] with ap and bp finite. The intersection of the
Kp is clearly the set of x E R such that
ap ::; x ::; bp for every pEl.
If we put a = sup ap and b = inf bp , the preceding relation is equivalent to
a ::; x ::; b by the definition of least upper bound and of greatest lower bound
of a family of real numbers. It remains to prove that the compact interval
[a, b], the intersection of the Kp as we have just seen, is nonempty, i.e. that
a ::; b.
But the fact that, for all p and q, the intersection Kp n Kq is nonempty
- it contains a Kr - shows that ap ::; bq • Since bq majorises all the ap , it
majorises their least upper bound a too. Since a minorises all the bq , it also
minorises their greatest lower bound b, whence a ::; b, qed. Compare with the
proof of Theorem 5 or, later, to the second proof of Theorem 13 (Cauchy's
criterion) .
Corollary 1 (Principle of nested intervals). Let It ::> 12 ::> ... be a
decreasing sequence of non empty compact intervals. Then the intersection of
the In is a nonempty compact interval.
Obvious; the set of indices in Theorem 10 is here N.
Corollary 2. Let (Ji)iEI be an infinite family of nonempty compact intervals. Then the intersection of Ji is nonempty if and only if for all
i l •...• in E I the intersection of the corresponding Ji is nonempty.
The condition of the statement is clearly necessary.
To show that it is sufficient let us put
18 The fact that we name the elements of I as p, q, r does not mean that we are
dealing with whole numbers: an arbitrary set is an arbitrary set.
225
§3. Bolzano-Weierstrass and Cauchy's criterion
9 - Nested intervals, Bolzano-Weierstrass, compact sets
The purpose of this nO is to prepare the proof of Cauchy's general criterion
of convergence that we will establish in the next n°. But these preliminary
results are, even so, also important.
It all rests on the following result:
Theorem 10. Let (Kp)PEI an arbitrary18 family of compact nonempty intervals. Suppose that, for all p, q E I, there exists an rEI such that
Kr C Kp n K q. Then the intersection of the Kp is a compact nonempty
interval.
Let us write Kp = lap, bp] with ap and bp finite. The intersection of the
Kp is clearly the set of x E R such that
ap ::; x ::; bp for every pEl.
If we put a = sup ap and b = inf bp , the preceding relation is equivalent to
a ::; x ::; b by the definition of least upper bound and of greatest lower bound
of a family of real numbers. It remains to prove that the compact interval
[a, b], the intersection of the Kp as we have just seen, is nonempty, i.e. that
a ::; b.
But the fact that, for all p and q, the intersection Kp n Kq is nonempty
- it contains a Kr - shows that ap ::; bq • Since bq majorises all the ap , it
majorises their least upper bound a too. Since a minorises all the bq , it also
minorises their greatest lower bound b, whence a ::; b, qed. Compare with the
proof of Theorem 5 or, later, to the second proof of Theorem 13 (Cauchy's
criterion) .
Corollary 1 (Principle of nested intervals). Let It ::> 12 ::> ... be a
decreasing sequence of non empty compact intervals. Then the intersection of
the In is a nonempty compact interval.
Obvious; the set of indices in Theorem 10 is here N.
Corollary 2. Let (Ji)iEI be an infinite family of nonempty compact intervals. Then the intersection of Ji is nonempty if and only if for all
i l •...• in E I the intersection of the corresponding Ji is nonempty.
The condition of the statement is clearly necessary.
To show that it is sufficient let us put
18 The fact that we name the elements of I as p, q, r does not mean that we are
dealing with whole numbers: an arbitrary set is an arbitrary set.