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 that there 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 we must 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.
Let ε>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