1. The Baire category theorem - Aarhus Universitet

Proof of Baire’s category theorem - continued
Finally the completion of the proof in the case of a complete metric
space:
As above B0 ⊇ B1 ⊇ B2 ⊇ B3 ⊇ ... and ∞ k=1 Bk ⊆ ∞ k=0 Vk ∩B0.
It remains therefore only to show that ∞ k=1 Bk = ∅.
Choose xk ∈ Bk. We claim that {xk} ∞ k=1 is a Cauchy sequence in S.
To see this, let ǫ > 0 be given. Choose N so large that 1
N ≤ ǫ.
Let n,m ≥ N. Then xn,xm ∈ BN, and hence d (xn,xm) ≤ 1
N ≤ ǫ.
Thus {xk} ∞ k=1 is a Cauchy sequence in S and we set
x = limk→∞ xk.
Since xn ∈ Bk when n ≥ k, we conclude that x ∈ Bk.
(Indeed, if x /∈ Bk there is a δ > 0 such that
c
{y ∈ S : d(y,x) < δ} ⊆ Bk . Since xn ∈ {y ∈ S : d(y,x) < δ}
for all large n, this is a contradiction.)
Hence x ∈ ∞ k=1 Bk and the proof is complete!
Klaus Thomsen 1. The Baire category theorem