Cauchy property of a series

Question

Are these two definitions equivalent, even though the first one has an extra term:

If we consider the series $\sum_{n=1}^{\infty}x_{n}$ and the formal definition of a Cauchy property defined in terms of the values $x_{n}$ as being:

1. For arbitrary $\epsilon > 0$ there exists a positive integer $N$ such that if $m > n > N$ we have $$|x_{n}+x_{n+1}+...+x_{m}| < \epsilon$$

is this equivalent to

2. For arbitrary $\epsilon > 0$ there exists a positive integer $N$ such that if $m > n > N$ we have $$|s_{n}-s_{m}| < \epsilon$$ where $|s_{n}-s_{m}| = |\sum_{k=m+1}^{n}x_{k}|$ are partial sums.

Are these equivalent and which one is the standard definition for the cauchy property of series in terms of $x_{n}$?

Answer 1

Hint: Try to see if you can deduce one definition from the other, that is, assuming that a sequence satisfies one definition, prove that it satisfies the other one.

Here is another example:

1. We say that $a_n \to \infty$ if for all $S$ there exists $N$ such that for all $n \geq N$, $a_n > S$.

2. We say that $a_n \to \infty$ if for all $S$ there exists $N$ such that for all $n > N$, $a_n > S$.

Answer 2

These are equivalent, $1.\Rightarrow 2.$ is a immediate consequence.

For $2.\Rightarrow 1.$ you can prove using the fact that $x_n\rightarrow 0$, so you can take a bigger $N$ such that summing $x_m$ for $m>N$ won't affect the inequality.