This is a major result which plays the role of the Heine Borel theorem for the set of
continuous functions. I will give the version which holds on an interval, although this theorem
holds in much more general settings. First is a definition of what it means for a collection of
functions to be equicontinuous. In words, this happens when they are all uniformly continuous
simultaneously.
Definition 6.11.1Let S ⊆ C
([0,T])
where C
([0,T])
denotes theset of functionswhich are continuous on the interval
[0,T]
. Thus S is a set of functions. Then S is said to beequicontinuous if whenever ε > 0 there exists a δ > 0 such that whenever f ∈ S and
|x − y|
< δ, it follows
|f (x) − f (y)| < ε
The set of functions is said to be uniformly bounded if there is a positive number M such thatfor all f ∈ S,
sup{|f (x)| : x ∈ [0,T]} ≤ M
Then the Ascoli Arzela theorem says the following.
Theorem 6.11.2Let
{f }
n
_{n=1}^{∞}⊆ C
([0,T ])
beuniformly bounded andequicontinuous. Then there exists a uniformly Cauchy subsequence.
Proof: Let ε > 0 be given and let δ correspond to ε∕4 in the definition of equicontinuity.
Let 0 = x_{0}< x_{1}<
⋅⋅⋅
< x_{n} = T where these points are uniformly spaced and the distance
between successive points is T∕n < δ. Then the points
{fn(x0)}
_{n=1}^{∞} is a bounded set in F.
By the Heine Borel theorem, there is a convergent subsequence
{ }
fk(0)(x0)
_{k(0)
=1}^{∞}. Thus
{k(0)}
denotes a strictly increasing sequence of integers. Then the same theorem implies
there is a convergent subsequence of this one, denoted as k
(1)
such that lim_{k(1)
→∞}f_{k(1)
}
(x0)
and lim_{k(1)
→∞}f_{k(1)
}
(x1)
both exist. Then take a subsequence of
{fk(1)}
called k
(2)
such that
for x_{i} = x_{0},x_{1},x_{2}, lim_{k(2)
→∞}f_{k(2)
}
(xi)
exists. This can be done because if a sequence
converges then every subsequence converges also. Continue this way. Denote by
{k}
the last
of these subsequences. Thus for each x_{i} of these equally spaced points of the interval,
lim_{k→∞}f_{k}
(xi)
converges. Thus there exists m such that if k,l ≥ m, then for each of these
x_{i},
|fk (xi)− fl(xi)| < ε
4
Let x ∈
[0,T]
be arbitrary. Then there is x_{i} such that x_{i}≤ x < x_{i+1}. Hence, for
k,l ≥ m,
This has shown that for every ε > 0 there exists a subsequence
{fk}
with the property
that
sup |f (x)− f (x)| < ε
x∈[0,T] k l
provided k,l are large enough. The argument also applies with no change to a given
subsequence in place of the original sequence of functions. That is, for any subsequence of the
original one, there is a further subsequence which satisfies the above condition. In what
follows
{fik}
_{k=1}^{∞} will denote a subsequence of
{f(i− 1)k}
_{k=1}^{∞}. Let ε_{i} = 1∕2^{i} so that
lim_{i→∞}ε_{i} = 0. Then let
{fik}
_{k=1}^{∞} denote a subsequence which corresponds to ε_{i} in the
above construction. Consider the following diagram.
The Cantor diagonal sequence is f_{k} = f_{kk} in the above. That is, it is the sequence
f11,f22,f33,f44,⋅⋅⋅
Then from the construction, f_{j},f_{j+1},f_{j+2},
⋅⋅⋅
is a subsequence of
{fjk}
_{k=1}^{∞}. Therefore,
there exists m such that k,l > m,
sup |fk(x)− fl(x)| < εj
x∈[0,T]
However, these ε_{j} converge to 0 and this shows that the diagonal sequence
{fj}
_{j=1}^{∞} just
described is a uniformly Cauchy sequence. ■
The process of obtaining this subsequence is called the Cantor diagonal process and occurs
in other situations.
From this follows an easy corollary.
Corollary 6.11.3Let
{fn}
_{n=1}^{∞}⊆ C
([0,T])
be uniformly bounded andequicontinuous. Then there exists a subsequence which converges uniformly to acontinuous function f defined on
[0,T]
.
Proof: From Theorem 6.9.11 the uniformly Cauchy subsequence from the Ascoli Arzela
theorem above converges uniformly to a function f. Now Theorem 6.9.7 this function f is also
continuous because, by this theorem, uniform convergence takes continuity with it and
imparts it to the limit function. ■
This theorem and corollary are major results in the theory of differential equations. There
are also infinite dimensional generalizations which have had great usefulness in the theory of
nonlinear partial differential equations.