Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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108 CHAPTER 16. MONTEL’S THEOREM<br />
• (fn,k+1) ∞ n=1 is a subsequence <strong>of</strong> (fn,k) ∞ n=1 , and<br />
• (fn,k(xk)) ∞ n=1 converges.<br />
For n ∈ N, set gn := fn,n. Then (gn) ∞ n=1 is a subsequence <strong>of</strong> (fn) ∞ n=1, and<br />
(gn(xk)) ∞ n=1 converges for each k ∈ N.<br />
We show that (gn) ∞ n=1 is a uniform Cauchy sequence on K (and thus convergent).<br />
Let ǫ > 0. Choose δ > 0 such that |f(x) − f(y)|< ǫ<br />
for all f ∈ F and for all<br />
3<br />
x,y ∈ K with |x−y|< δ. Since K is compact, there exist y1,...,yν ∈ K such that<br />
K ⊂ �ν j=1Bδ (yj). Since {x1,x2,x3,...} is dense in K, there exist k1,...,kν ∈ N such<br />
2<br />
that xkj ∈ Bδ(yj).<br />
It follows that K ⊂<br />
2<br />
�ν Bδ(xkj j=1 ).<br />
By construction, (gn(xk)) ∞ n=1 is a Cauchy sequence for each k ∈ N. Choose N ∈ N<br />
such that<br />
ǫ<br />
|gn(xkj )−gm(xkj )|<<br />
3<br />
for n,m ≥ N and j = 1,...,ν. Let x ∈ K be arbitrary, and let n,m ≥ N. Choose<br />
j ∈ {1,...,ν} such that x ∈ Bδ(xkj ), and note that<br />
|gn(x)−gm(x)|≤ |gn(x)−gn(xkj )| +|gn(xkj )−gm(xkj )|<br />
Hence, (gn) ∞ n=1<br />
� �� �<br />
< ǫ<br />
3<br />
� �� �<br />
< ǫ<br />
3<br />
is a uniform Cauchy sequence on K.<br />
+|gm(xkj )−gm(x)|<br />
� �� �<br />
< ǫ<br />
< ǫ.<br />
3<br />
Proposition 16.1. Let D ⊂ R N be open, and let F be a family <strong>of</strong> functions from D<br />
to R M that is equicontinuous and uniformly bounded on compact subsets <strong>of</strong> D. Then<br />
every sequence in F has a compactly convergent subsequence.<br />
Pro<strong>of</strong>. For each k ∈ N, define Kk := Bk[0] if D = R N and Kk := Bk[0] ∩{x ∈ D :<br />
dist(x,∂D) ≥ 1<br />
k } if D �= RN . Notice that<br />
• � ∞<br />
k=1 Kk = D and<br />
• Kk ⊂ intKk+1 for n ∈ N.<br />
Let (fn) ∞ n=1 be a sequence in F. By the Arzelà–Ascoli Theorem, there exists a<br />
subsequence (fn,1) ∞ n=1 <strong>of</strong> (fn) ∞ n=1 and a function g1: K1 → RM such that fn,1|K1→ g1<br />
uniformlyonK1. Invoking theArzelà–AscoliTheoremagain,weobtainasubsequence<br />
(fn,2) ∞ n=1 <strong>of</strong> (fn,1) ∞ n=1 and a function g2: K2 → RM such that fn,2|K2→ g2 uniformly<br />
on K2. Inductively, we thus obtain, for each k ∈ N, a subsequence (fn,k) ∞ n=1 <strong>of</strong> (fn) ∞ n=1<br />
and a function gk: Kk → RM such that, for each k ∈ N,<br />
• (fn,k+1) ∞ n=1 is a subsequence <strong>of</strong> (fn,k) ∞ n=1 , and<br />
• fn,k|Kk → gk uniformly on Kk.