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Appendix to Chapter III
pairwise disjoint arcs. If one defines the distance between two points of C in
the usual way one then sees that C has no connected neighbourhoods.
3 - Limits and Cauchy's criterion in a metric space; complete spaces
Everything we said in nO 5 of Chap. II about scalar sequences extends without
modification to sequences of points of a Cartesian or metric space X: it
suffices to replace expressions of the form Ix - yl everywhere by the distance
function d(x, y), which we have already often done to accustom the reader.
The concept of limit of a sequence is thus defined by the relation
(3.1)
which brings one back to limits in lR. One sees immediately that
(3.2)
limxn = a & limYn = b =? limd(xn,Yn) = d(a,b)
on using the triangle inequality (or, in this particular case, the quadrilateral
... ).
If E is a subset of X the adherent points of E are clearly the limits of
convergent sequences of points of E. It follows that E is closed if and only if
every limit of points of E belongs to E.
If for example X is the space B(M) of functions M -----+ bounded on a set M, with the distance dM(f, g) = Ilf - gliM' then convergence of a sequence (fn) to a limit f is precisely uniform convergence on M,
since, for all r > 0, the relation dM(f,g) :::; r means that If(x) - g(x)1 :::; r
for all x EM. When M c see below], one can consider the subset CO(M) of B(M) formed by the continuous scalar functions on M. Theorem 8 of Chap. III, nO 5, says that if a
sequence of functions fn E CO(M) converges uniformly on M, i.e. converges
in the metric space B(M), then the limit function is again continuous, i.e.
still belongs to CO(M). Conclusion: CO(M) is a closed subset of the metric
space B(M). If M is a compact interval in JR, the set E = CP(M) of functions of class CP in M is not closed in X = CO(M), since a uniform limit
of differentiable functions is not necessarily differentiable. But Theorem 27
of Chap. V, nO 27, will show that every element of X is a limit of elements
of E (one can even restrict oneself to considering polynomial functions, by
Weierstrass' theorem). When every element of a metric space X is the limit
of elements of a given set E c X, i.e. when E = X, one says that E is everywhere dense in X, a concept introduced by Cantor for X = JRj for example,
Q is everywhere dense in lR.
The definition of a limit extends no less immediately to scalar functions
(i.e. with values in to b E 0, there exists an r' > 0
such that