If $M$ is an arbitrary metric space, the following holds:
$A\subseteq M$ is totally bounded $\Leftrightarrow$ Each sequence in $A$ contains a Cauchy subsequence.
Additionally, for $M=\mathbb{R}^n$ with the Euclidean metric, the following is true:
$A\subseteq M$ is totally bounded $\Leftrightarrow$ $A\subseteq M$ is bounded.
In short: Total boundedness can be characterized by Cauchy sequences; and in Euclidean spaces, total boundedness and boundedness are the same.
I am looking for a sequence property (let's call it $P$ for brevity), such that the following two statements be true:
If $M$ is an arbitrary metric space, then:
$A\subseteq M$ is bounded $\Leftrightarrow$ Each sequence in $A$ contains a subsequence with property $P$.
If $M=\mathbb{R}^n$ and $(x_n)_{n\in\mathbb{N}}$ is a sequence in $M$, then:
$(x_n)_{n\in\mathbb{N}}$ has property $P$ $\Leftrightarrow$ $(x_n)_{n\in\mathbb{N}}$ is a Cauchy sequence.
In short: Boundedness should be characterizable by $P$ sequences and in Euclidean spaces $P$ sequences and Cauchy sequences should be the same.
My question:
What property should $P$ be, such that the above holds?
My thoughts and what I have done so far:
I think the following definition would do:
$(x_n)_{n\in\mathbb{N}}$ has property $P$
$:\Leftrightarrow$ $(x_n)_{n\in\mathbb{N}}$ is bounded and all Cauchy subsequences of $(x_n)_{n\in\mathbb{N}}$ are equivalent.
(Figuratively, this means that a $P$ sequence has at most one accumulation point. However, since the metric space need not be complete, instead of accumulation points I use the equivalence of Cauchy sequences.)
Then:
- If $(x_n)_{n\in\mathbb{N}}$ is a bounded sequence which contains no Cauchy subsequence, then $(x_n)_{n\in\mathbb{N}}$ itself has property $P$ (because if there isn't a Cauchy subsequence all Cauchy subsequences are trivially equivalent).
- If $(x_n)_{n\in\mathbb{N}}$ is a bounded sequence which does contain a Cauchy subsequence, then this Cauchy subsequence has property $P$ (because all subsequences of a Cauchy sequence are equivalent Cauchy sequences).
- If every sequence in $A$ contains a subsequence with property $P$, then $A$ must be bounded or else an unbounded divergent sequence could be constructed which does not contain a subsequence with property $P$.
So: $A$ is bounded iff each sequence in $A$ contains a subsequence with property $P$.
Additionally, we have:
- In an arbitrary metric space every Cauchy sequence has property $P$.
- In $\mathbb{R}^n$ every bounded sequence and thus every $P$ sequence contains a Cauchy subsequence. One can show that then a $P$ sequence is a Cauchy sequence.
So: In $\mathbb{R}^n$, Cauchy sequences and $P$ sequences coincide.
I conclude that my definition for the property $P$ seems to work. However, I don't find this definition esthetically nice and I wonder if the $P$ property can be defined in a prettier way.
I don't think there will be a nice property. Consider a space $X$ with the discrete metric. Then $X$ is bounded, but every sequence in $X$ is either divergent or eventually constant.