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Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?
I was thinking about sequences where it appears the terms get closer and closer together, and wondered if they converge.
Now let's first define a few things. When I say "the terms get closer and closer together", I mean "the distance between any two consecutive terms approaches zero." In other words, for a sequence $\left(x_n\right)$,
$$|x_n-x_{n-1}| \to 0$$
consecutive terms become closer and closer together.
Let's look at an example: $\left(\ln n\right)$. Clearly,
$$\bigl|\ln (n) - \ln (n-1)\bigr|\to 0$$
and this can be verified by looking at a graph. At first, I saw this and thought $\left(x_n\right)$ and $\left(\ln n\right)$ looked like Cauchy sequences, and this was bugging me for the longest time, because I knew $\left(\ln n\right)$ was not supposed to be Cauchy! But I realize now that there is a subtle difference: for a Cauchy sequence $\left(y_n\right)$,
$$|y_n-y_{m}| \to 0$$
So in the cases of $\left(x_n\right)$ and $\left(\ln n\right)$, it may be true that consecutive terms are closer together, but two arbitrary terms aren't necessarily close together. So $\left(\ln n\right)$ is definitely not Cauchy.
What can we call sequences such as $\left(x_n\right)$ and $\left(\ln n\right)$, where consecutive terms become closer together? I'd like to propose a name: let's call them Cauchy-ish.
An intuitive geometric view of a Cauchy-ish sequence $\left(x_n\right)$ might be that you have a bunch of points on a line, and as you move forward in the sequence, the points get closer and closer together. It seems to me that this sequence would converge, no? Obviously my intuitive side and my analytical side disagree, because $\left(\ln n\right)$ is Cauchy-ish but is not Cauchy.
My question, finally, is, why don't "Cauchy-ish" sequences necessarily converge?