proof of Weierstrass M-test

Consider the sequence of partial sums sn=m=1nfm. Take any p,q such that pq,then, for every xX, we have

|sq(x)-sp(x)| = |m=p+1qfm(x)|

But since n=1Mn converges, for any ϵ>0 we can find an N such that, for any p,q>N and xX, we have |sq(x)-sp(x)|m=p+1qMm<ϵ. Hence the sequence sn converges uniformly to n=1fn.

Title proof of Weierstrass M-test
Canonical name ProofOfWeierstrassMtest
Date of creation 2013-03-22 12:58:01
Last modified on 2013-03-22 12:58:01
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Proof
Classification msc 30A99
Related topic CauchySequence