108 CHAPTER 6. MULTI-VARIABLE CALCULUS

Theorem 6.7.9 If a sequence {an} in Fp converges, then the sequence is a Cauchy se-quence.

Proof: Let ε > 0 be given and suppose an→ a. Then from the definition of convergence,there exists nε such that if n > nε , it follows that

|an−a|< ε

2

Therefore, if m,n≥ nε +1, it follows that

|an−am| ≤ |an−a|+ |a−am|<ε

2+

ε

2= ε

showing that, since ε > 0 is arbitrary, {an} is a Cauchy sequence.

6.7.2 Continuity And The Limit Of A SequenceJust as in the case of a function of one variable, there is a very useful way of thinking ofcontinuity in terms of limits of sequences found in the following theorem. In words, it saysa function is continuous if it takes convergent sequences to convergent sequences wheneverpossible.

Theorem 6.7.10 A function f : D(f)→ Fq is continuous at x∈D(f) if and only if, wheneverxn→ x with xn ∈ D(f) , it follows f(xn)→ f(x) .

Proof: Suppose first that f is continuous at x and let xn→ x. Let ε > 0 be given. Bycontinuity, there exists δ > 0 such that if |y−x|< δ , then |f(x)− f(y)|< ε. However, thereexists nδ such that if n≥ nδ , then |xn−x|< δ and so for all n this large,

|f(x)−f(xn)|< ε

which shows f(xn)→ f(x) .Now suppose the condition about taking convergent sequences to convergent sequences

holds at x. Suppose f fails to be continuous at x. Then there exists ε > 0 and xn ∈ D( f )such that |x−xn|< 1

n , yet|f(x)−f(xn)| ≥ ε.

But this is clearly a contradiction because, although xn→ x, f(xn) fails to converge to f(x) .It follows f must be continuous after all. This proves the theorem.

6.8 Properties Of Continuous FunctionsFunctions of p variables have many of the same properties as functions of one variable.First there is a version of the extreme value theorem generalizing the one dimensional case.

Theorem 6.8.1 Let C be closed and bounded and let f : C→ R be continuous. Then fachieves its maximum and its minimum on C. This means there exist, x1,x2 ∈C such thatfor all x ∈C,

f (x1)≤ f (x)≤ f (x2) .