Proof
Let be the limit of the sum . Since the sum is absolutely convergent, call its limit .
By convergence of the sum, for there exists an integer such that . We will show that converges uniformly by showing that . The crucial point here is that does not depend on .
Read more about this topic: Weierstrass M-test
Famous quotes containing the word proof:
“A short letter to a distant friend is, in my opinion, an insult like that of a slight bow or cursory salutation—a proof of unwillingness to do much, even where there is a necessity of doing something.”
—Samuel Johnson (1709–1784)
“War is a beastly business, it is true, but one proof we are human is our ability to learn, even from it, how better to exist.”
—M.F.K. Fisher (1908–1992)
“It comes to pass oft that a terrible oath, with a swaggering accent sharply twanged off, gives manhood more approbation than ever proof itself would have earned him.”
—William Shakespeare (1564–1616)