The following proof is a rather detailed one.

Let c[a,b] denote the space of continuous ,real valued function, over [a,b].

To show that c[a,b] is complete w.r.t the supnorm we must show that:

Given a cauchy sequence of continuous functions over [a,b] we must show that ,there exists a function ,g belonging to c[a,b] and w.r.t the supnorm.

SO

Let { } be a cauchy sequence

Let xε[a,b]

Let ε>0

SINCE { }is cauchy in c[a,b] there exists N belonging to natural Nos and such that:

for all n.m: and ====> sup{ :xε[a,b]}< ε.

Because ε{ :xε[a,b]}======> <ε

Hence.

{for all ε>0 there exists NεN AND SUCH THAT ,for all n,m: =====> }................................................. ...............................1

Now (1)====> { } is a cauchy sequence in real Nos R ====> THERE EXISTS a unique yεR AND

Thus

for all x :xε[a,b]====>THERE EXISTS a unique yεR AND

That defines a function g:[a,b]------->R and such that:

for all x:xε[a,b]======> .................................................. .......................................2

So up to now we have proved thatthere exists ,g:[a,b]------>R.

Now we need to show that gεc[a,b] i.e g is continuous over [a,b]

Since we have the theorem that states :uniform convergence of continuous functions to a function implies that the function is continuous wemust show uniform convergence.

Let ε>0

by (1) there exists NεN AND such that:

for all n,m: =====> <ε/2................................................. ...........................

and if and xε[a,b] by (2) we have ======>g(x) is an accumulation point=======>there exists and <ε/2

and hence <ε/2 and <ε/2 ====> <ε (by the triangular inequality) .

Τherefor { } converges uniformly to g and so gεc[a,b]

Sofar we have proved:there exists a gεc[a,b].

So it remains the proof that { } converges to g w.r.t the supnorm.

Letε>0

by uniform convergence ,there exists NεN AND such that ,

for all n,x: and xε[a,b]======> <ε/2.

Let

because yε{ :xε[a,b]}======>y<ε/2=====> the set { :xε[a,b]} is bounded above by ε/2=====>sup{ :xε[a.b]} <ε

AND so c[a,b] is complete w.r.t the supnorm