3.3. THE CASE OF A CLOSED SET IN Rp 83

Proof: Let { fn} be a Cauchy sequence of functions in C0 (X). Then in particular,{ fn (x)}∞

n=1 is a Cauchy sequence in F. Let f (x)≡ limn→∞ fn (x) .Let ε > 0 be given. Thenthere exists N such that for any x ∈ X ,

| fm (x)− fn (x)| ≤ ∥ fm− fn∥∞< ε/3

for m,n≥ N. Thus, picking n≥ N and taking a limit as m→∞, | f (x)− fn (x)| ≤ ε/3 sincex was arbitrary,

supx∈X| f (x)− fn (x)| ≤ ε/3 (3.2)

By assumption, there exists a compact set K such that if x /∈ K then ∥ fN∥∞< ε/3. Thus,

from 3.2,

supx/∈K| f (x)| ≤ 2ε/3 < ε

It remains to verify that f is continuous. Letting N be as the above, let x,y ∈ X . Then

| f (x)− f (y)| ≤ | f (x)− fN (x)|+ | fN (x)− fN (y)|+ | fN (y)− f (y)|

By continuity of fN at x, there exists δ > 0 such that if |x− y|< δ for y ∈ X , it follows that| fN (x)− fN (y)|< ε/3. Then for |y− x|< δ ,

| f (x)− f (y)| ≤ supx∈X| f (x)− fN (x)|+ ε

3+ sup

y∈X| f (y)− fN (y)|

<ε

3+

ε

3+

ε

3= ε

showing that f is continuous. Thus the sequence of functions converges uniformly to afunction f ∈C0 (X) which is what it means to be complete. Every Cauchy sequence con-verges. Indeed 3.2 says that ∥ f − fn∥∞

< ε for f ∈C0 (X). ■The above refers to functions which have values in C but the same proof works for

functions which have values in any complete normed linear space.In the case where the functions in C0 (X) all have real values, I will denote the resulting

space by C0 (X ;R) with similar meanings in other cases.The following has to do with a trick which will enable a result valid on sets which are

only closed rather than compact. In general, you consider a locally compact Hausdorffspace instead of a closed subset of Rp. Consider the unit sphere in Rp+1, centered at thepoint (0, · · · ,0,1)≡

(⃗0,1)

.

Sp ≡

{⃗x ∈ Rp+1 : (xn+1−1)2 +

n

∑k=1

x2k = 1

}

Define a map from Rp which is identified with Rp×{0} to the surface of this sphere asfollows. Extend a line from the point, p⃗ in Rp to the point

(⃗0,2)

on the top of this sphereand let θ (p) denote the point of this sphere which the line intersects.