10. Commutative Banach algebras - Aarhus Universitet

Quotients
Proof.
We continue by induction to construct a sequence {xj} ∞ j=1
such that
a) q (xj) = zj = yNj ,
b) xj − xj+1 < 2 −j ,
in X
for all j.
It follows from b) that xm − xn ≤
xm − xm−1 + xm−1 − xm−2 + · · · + xn+1 − xn ≤ m−1
j=n 2−j .
Thus {xj} is a Cauchy sequence in X and x = limj→∞ xj exists in
X because X is complete.
We finish the proof by checking that y = q(x) is the limit {yn} in
X/J:
Klaus Thomsen 10. Commutative Banach algebras