110 CHAPTER 4. CONTINUITY AND LIMITS

Theorem 4.9.3 Let fn be continuous and each fn bounded on D and suppose thatlimn→∞ ∥ fn − f∥= 0. Then f is also continuous. If each fn is uniformly continuous, then fis uniformly continuous.

Proof: Let ε > 0 be given and let x ∈ D. Let n be such that ∥ fn − f∥< ε

3 . By continuityof fn there exists δ > 0 such that if |y− x|< δ , then | fn (y)− fn (x)|< ε

3 . Then for such y,

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

<ε

3+∥ f − fn∥+∥ fn − f∥< ε

3+

ε

3+

ε

3= ε

and so this shows that f is continuous. To show the claim about uniform continuity, use thesame string of inequalities above where δ is chosen so that for any pair x,y with |x− y| <δ , | fn (y)− fn (x)| < ε

3 . Then the above shows that if |x− y| < δ , then | f (x)− f (y)| < ε

which satisfies the definition of uniformly continuous.This implies the following interesting corollary about a uniformly Cauchy sequence of

continuous functions.

Definition 4.9.4 Let { fn} be a sequence of continuous functions defined on [a,b]. Itis said to be uniformly Cauchy if for every ε > 0 there exists nε such that if m,k > nε , then∥ fm − fk∥< ε .

Corollary 4.9.5 Suppose { fn} is a uniformly Cauchy sequence of continuous uniformlybounded functions defined on D. Then there exists a unique continuous function f such that

limn→∞

∥ fn − f∥= 0

If each fn is uniformly continuous, then so is f .

Proof: The hypothesis implies that { fn (x)} is a Cauchy sequence in R for each x.Therefore, by completeness of R, Theorem 3.7.3, this sequence converges for each x. Letf (x)≡ limn→∞ fn (x). Then for m > n,

| f (x)− fn (x)| ≤ supm

| fm (x)− fn (x)| ≤ supm

∥ fm − fn∥< ε

provided n is sufficiently large. Since x is arbitrary, it follows that ∥ f − fn∥ ≤ ε for all nlarge enough which shows by definition that limn→∞ ∥ fn − f∥= 0.

Now the continuity of f follows from Theorem 4.9.3 and if each fn is uniformly con-tinuous, then so is f . How many such functions f are there? There can be only one becausef (x) must equal the limit of fn (x).

4.10 Polynomials and Continuous FunctionsIt turns out that if f is a continuous real valued function defined on an interval, [a,b] thenthere exists a sequence of polynomials, {pn} such that the sequence converges uniformly tof on [a,b]. I will first show this is true for the interval [0,1] and then verify it is true on anyclosed and bounded interval. First here is a little lemma which is interesting in probability.It is actually an estimate for the variance of a binomial distribution.