§2. Uniform convergence
211
li(x) - Pn(x)1 < lin for all x E K and all n 2:: k,
so that the sequence (Pn) converges uniformly to i on K, qed.
Figure 8 shows an example of compact convergence on lR where the in do
not converge uniformly on R
fig. 8.
6 - A slip up of Cauchy's
It is interesting to see 7 that Cauchy, attempting to ground analysis on a
rigorous basis, nevertheless "proved" in his COUTS d'analyse (1821) at the
Ecole polytechnique, that every simple limit of continuous functions is again
continuous. Figure 9 "confirms" the "theorem": the limit function equals 0
for x = 0 and 1 for x > O. In fact, Cauchy dealt with a series of continuous
functions, but this amounts to the same since he introduced the total sum s(x)
of the series, its partial sums sn(x) and the "remainder" Tn(X) = s(x)-sn(x).
He argued as follows:
fig. 9.
7 For all that follows, see Paul Dugac, Sur les fondements de l'Analyse de Cauchy
a Baire (doctoral thesis, Universite Pierre et Marie Curie, 1978).