Prove a metric space is totally bounded iff it is bounded in every equivalent metric. Prove a metric space is totally bounded iff it is bounded in every equivalent metric.
I was able to see the solution to this problem if we change the wording to "iff it is totally bounded in every equivalent metric." I was wondering whether there is a typo in the wording because bounded does not imply totally bounded, and just because the metric space is bounded in equivalent metric does not give much information as to whether space is totally bounded.
 A: This is false. $(0,1)$ is totally bounded w.r.t. the usual metric. An equivalent metric  is $|\frac 1  x-\frac 1 y|$ and $(0,1)$ is not bounded in this metric.
[Definition of equivalent metrics I am using: two metrics are equivalent if they have the same convergent sequeneces with the same limits; equivalently, they have the same open sets].
A: My guess about this problem is that there are two issues:

*

*There was a typo and the problem should read


Prove a metric space is totally bounded iff it is totally bounded in every equivalent metric.

and


*The definition of "equivalent" in the problem was not completely standard, the author of the problem was assuming strong equivalence of metrics. (Or, more generally, that the identity map between the two metric spaces $(X,d_1)\to (X,d_2)$ is a uniform homeomorphism.)

With these two corrections, the problem becomes a pleasant exercise.
Addendum. Another possibility is that the question was meant to be:

Prove that a metric space is totally bounded if and only if it is bounded in every uniformly equivalent metric.

Here two metrics $d_1, d_2$ on a set $X$ are called uniformly equivalent if both identity maps
$$
id: (X, d_1)\to (X,d_2), id: (X, d_2)\to (X,d_1)
$$
are uniformly continuous.
This question again has a positive answer but a proof requires much more work. Since the question is likely used as a homework or an exam problem, I will refrain from writing a proof.
