Why is Cauchy condition for convergence not formulated in a simpler way?

Question

The standard definition of a Cauchy sequence (e.g. it's given in Wikipedia and most textbooks I remember; admittedly those are mostly older ones) is:

for every positive real $ε > 0$ there is a positive integer $N$ such that for all integers $m,n > N$, $|x_m - x_n| < \varepsilon$.

However, this is easily seen to be equivalent to

for every positive real $ε > 0$ there is a positive integer $N$ such that for all positive integers $m > N$, $|x_m - x_N| < \varepsilon$.

(Just take $\varepsilon$ in this definition to be half of the one from the standard definition.)

Why isn't this simpler definition more commonly used? My best hypotheses are:

1. That's the formulation Cauchy used and so it remains, since it's only a slight simplification.

2. Authors prefer the symmetry between $n$ and $m$ to simplified proofs.

3. It doesn't generalize to some settings (which ones?).

4. There are actually lots of textbooks I haven't seen which use this definition.

Answer 1

Despite having one more variable, I think the standard definition is conceptually simpler (it says that as $m \to \infty$ and $n \to \infty$, the distance from $x_m$ to $x_n$ goes to $0$).

Answer 2

You can make it even more general. The formal definition of convergence in a metric space $(M,d)$ is

$$∃ a ∈ M\; ∀ ε>0 \;∃ N ∈ \Bbb N \;∀ n ≥ N: d(x_n,a)<ε.$$

By exchanging the first two quantifier blocks, you get a formula that is equivalent to the Cauchy criterion,

$$∀ ε>0 \;∃ a ∈ M\; ∃ N ∈ \Bbb N \;∀ n ≥ N: d(x_n,a)<ε.$$

Answer 3

In line with Robert Israel's response: the two may be logically equivalent, but conceptually they represent different ideas. The standard definition says that any two elements of the sequence with sufficiently large index are close to each other. Your definition says that all elements of the sequence with sufficiently large index are close to a specific element of the sequence, which depends on how close we're talking. I think the concept behind the standard definition lends itself better to the intuition that unless there's a point missing in the underlying space, the elements of the sequence should also be close to another point, the limit of the Cauchy sequence.