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Corollary 3.0.4 If f : [a,b]→ R is continuous, then there exists xM ∈ [a,b] such that

f (xM) = sup{ f (x) : x ∈ [a,b]}

and there exists xm ∈ [a,b] such that

f (xm) = inf{ f (x) : x ∈ [a,b]}

Proof: From the definition of inf{ f (x) : x ∈ [a,b]} , there exists xn ∈ [a,b] such that

f (xn)≤ inf{ f (x) : x ∈ [a,b]}+1/n

That is, limn→∞ f (xn)= inf{ f (x) : x ∈ [a,b]} . This is called a minimizing sequence. There-fore, there is a subsequence

{xnk

}which converges to x ∈ [a,b] . By continuity of f it

follows thatinf{ f (x) : x ∈ [a,b]}= lim

k→∞f(xnk

)= f (x)

The case where f achieves its maximum is similar. You just use a maximizing sequence.■

Corollary 3.0.5 If {xn} is a Cauchy sequence, then it converges.

Proof: The Cauchy sequence is contained in some closed interval [a,b]. This is be-cause, letting ε = 1, it follows that there exists N such that if m,n≥N, then |xn− xm|< 1. Inparticular, for all n≥ N, |xn− xN |< 1. Therefore, |xn| ≤max{|xN |+1, |x1| , |x2| , · · · , |xN |}for all n. By Corollary 3.0.3, there is a subsequence of the Cauchy sequence, denoted as{

xnk

}which converges to some x ∈ [a,b]. Since the original sequence is a Cauchy se-

quence, letting ε > 0 be given, there is N such that if k, l ≥ N, then |xk− xl | < ε/2 and∣∣xnk − x∣∣< ε/2. Thus if m≥ N, then

|x− xm| ≤ |x− xnm |+ |xnm − xm|<ε

2+

ε

2= ε

Indeed, if m ≥ N, then nm ≥ N because {xnm}∞

m=1 is a subsequence. Thus the originalCauchy sequence converges to x. ■

Actually, the convergence of every Cauchy sequence is equivalent to completeness andso it gives another way of defining completeness in contexts where no order is available.Recall completeness means that every nonempty set bounded above (below) has a leastupper bound (greatest lower bound). More consideration of this issue is a good topic foradvanced calculus courses. This standard definition involving least upper bounds dependson an order. One can prove that if you have the least upper bound property described in theabove definition, then you also have the greatest lower bound property also described thereand the other way around.

The Riemann integral pertains to bounded functions which are defined on a boundedinterval. Let [a,b] be a closed interval. A set of points in [a,b], {x0, · · · ,xn} is a partition if

a = x0 < x1 < · · ·< xn = b.

Such partitions are denoted by P or Q.