Convergence and Cauchy Criterion

1. Sep 24, 2015

andyfeynman

1. The problem statement, all variables and given/known data
Suppose the sequence (xn) satisfies |xn + 1 - xn| < 1/n2, prove that (xn) is convergent.

2. Relevant equations
|xn - xm| < ɛ

3. The attempt at a solution
If m > n, then
|xn - xm|
< |xn - xn + 1| + |xn + 1 - xn + 2| + ... + |xm - 1 - xm|
< 1/n2 + 1/(n+1)2 + ... + 1/(m - 1)2
< [1/(n - 1) - 1/n] + [1/n - 1/(n + 1)] + ... + [1/(m - 2) - 1/(m - 1)] = 1/(n - 1) - 1/(m - 1)
< 1/(n - 1)

Let ɛ be given. Choose m > n > N := [1/ɛ] + 1 such that |xn - xm| < ɛ for all m > n > N.

Is there any problem with my proof?

2. Sep 24, 2015

Fredrik

Staff Emeritus
The calculations look good, but the statements surrounding it do not. In particular:
1. It doesn't make sense to say that you choose m such that some statement is true for all m.
2. You didn't say that $\varepsilon$ is a positive real number, so the reader can wonder how you intend to make $|x_n-x_m|<\varepsilon$ when $\varepsilon=-1$.
3. The calculation didn't actually conclude that $|x_n-x_m|<\varepsilon$.
Edit: 4. If the [x] notation means what I think it does, your definition of N doesn't work.

Last edited: Sep 24, 2015