Convergence and Cauchy Criterion

[andyfeynman]

Homework Statement

Suppose the sequence (x_n) satisfies |x_{n + 1} - x_n| < 1/n², prove that (x_n) is convergent.

Homework Equations

|x_n - x_m| < ɛ

The Attempt at a Solution

If m > n, then
|x_n - x_m|
< |x_n - x_{n + 1}| + |x_{n + 1} - x_{n + 2}| + ... + |x_{m - 1} - x_m|
< 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 |x_n - x_m| < ɛ for all m > n > N.

Is there any problem with my proof?

 

[Fredrik]

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.