TITLE: Definition of Cauchy Sequence
QUESTION [9 upvotes]: I have a question regarding the definition of a Cauchy sequence of a sequence in a metric space.
The definition I learned and that is consistent with Wikipedia defines
a sequence $(x_n)_{n=1}^\infty$ as a Cauchy sequence if
$$
\forall\, \varepsilon>0 \;\;\exists\, N\in\mathbb{N}\;\;
\forall\, m,n \geq N : d(x_m,x_n)<\varepsilon
$$
If I am not mistaken, there is a simpler, but equivalent definition:
$$
\forall\, \varepsilon>0 \;\; \exists\, N\in\mathbb{N} \;\;
\forall\, m \geq N: d(x_m,x_N)<\varepsilon
$$
This is simpler, because it only has two natural numbers in it instead of three.
This makes it easier to prove, that a given sequence is a Cauchy sequence.
Note that the equivalence relies on the triangle inequality.
Proof: $(\Rightarrow)$: we simply choose $n=N$.
$(\Leftarrow)$: Let $\varepsilon>0$.
Then
$$
\exists\, N\in\mathbb{N}\;\;
\forall\, m \geq N: d(x_m,x_N)<\frac12\varepsilon
$$
This means that for $m,n\geq N$ we have
$$
d(x_m,x_n) \leq d(x_m,x_N)+d(x_n,x_N)
< \frac12\varepsilon +\frac12\varepsilon
= \varepsilon
$$
So here is my question: why did I never encounter the more simple definition before?
Did I make a mistake somewhere? Are there advantages to the common definition, that I don't see?
Edit: Often the shortest/simplest definition becomes the standard definition. Why not in this case?

REPLY [2 votes]: The alternative definition is not simpler than the standard one. This is because theorems are complicated by the alternative definition, the alternative definition is not generalizable and it is not intuitive.
Here some cases that will convince you of this.

Conterintuitive. The standard definition translates in "a sequence $x_n$ has elements arbitrarily close to each other for sufficiently large $n$." With the alternate definition it is a bit harder to have this intuition.
Conterintuitive. Form the standard definition it is immediate to deduce the following equivalent definition for a fundamental sequence
$$\lim\limits_{n\to\infty\\m\to\infty}d(x_n,x_m)=0$$
Difficult to use. A fundamental sequence $x_n$ having a subsequence $x_{n_k}$ converging to a limit converges to that same limit.
Indeed it results that
$$\forall n_k>N, d(x_n,x_{n_k})<\varepsilon$$
Taking the limit: $d(x_n, l)<\varepsilon$, that is, $x_n$ converges to $l$.
It would be impratical to prove this using the alternate definition.
Conterintuitive. Fundamental sequences are "fundamental" to complete a metric space. Points of the completed space are identified by fundamental sequences of points of the original space. 
Two fundamental sequences $a_n$ and $b_n$ are equivalent when $$\forall \varepsilon>0, \exists N>0, \forall n>N, d(a_n,b_n)<\varepsilon$$
From the standard definition of fundamental sequence it is immediate to see that all subsequences of a fundamental sequence are equivalent one with each other. The alternative definition is impratical for the purpose.
Difficult to use. Standard definition in metric spaces fits well with definition in other spaces.
For instance, in the rationals standard and your alternative definitions translate into:

$$\forall \varepsilon>0,\exists N>0, \forall n,m>N, |x_n-x_m|<\varepsilon$$
$$\forall \varepsilon>0,\exists N>0, \forall n>N, |x_n-x_N|<\varepsilon$$
In $\mathbb{Q}$ given two fundamental sequences $a_n$ and $b_n$ their sum $a_n+b_n$ and product $a_nb_n$ are fundamental.
Thanks to these two inequalities:
$$|a_n+b_n-a_m-b_m|\le|a_n-a_m|+|b_n-b_m|$$
$$|a_nb_n-a_mb_m|=|a_nb_n-a_mb_n+a_mb_n-a_mb_m|\le|b_n||a_n-a_m|+|a_m||b_n-b_m|$$
the proof follows straightforwardly from the standard definition of fundamental sequence. The alternative definition is impratical for the purpose.

Not generalizable. The alternative definition cannot be put for fundamental nets in topological linear spaces. Given a net $x_{\alpha}$, this net is fundamental if the net $x_{\alpha'} - x_{\alpha''}$ converges to $0$. This is, in a way, parallel to the standard definition of fundamental sequence of metric space. The alternate definition does not have a counterpart in topological vector space.

The list goes on.