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On page 47 of ''A Course in Metric Geometry'' by Burago, Burago and Ivanov there is a version of Arzela-Ascoli theorem:

In a compact metric space, any sequence of curves with uniformly bounded lengths contains a uniformly converging subsequence.

In the proof they write "For each $\gamma_i$, there is a unique constant speed parameterization by the unit interval $[0,1]$. Uniform boundedness of the lengths of $\gamma_i$ means that the speeds of these parameterizations are uniformly bounded.".

Is this supposed to be their definition of the lengths being uniformly bounded or is this just a consequence? If this is not the definition then I don't understand what uniformly bounded lengths means. How is it any different from saying the lengths are bounded? Are they using the word uniformly just to emphasize they are bounded by the same constant as opposed to just all being finite?

The book is available from http://www.math.psu.edu/petrunin/papers/alexandrov/bbi.pdf.

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In my opinion this is a rather trivial statement (with all the details which arise from the rather general setup to be carefully kept in mind)

Uniformly bounded length means of course that there is $L_0>0$ such that the length of the $\gamma_i$ is bounded from above uniformly (i.e. independent of $i$) by $L_0$. This statement does not involve any parametrization. Now in order to get your hands on something you need to be able to compare the curves. For this they want to parametrize the curves on the same fixed interval (e.g. $[0,1]$). In order to acchieve that they use constant speed parametrization, but with each curve having possibly a different (constant) speed.

Now if constant speed $v$ means that you travel a distance of $v$ in a period of time of length $1$ and that this scales (i.e you travel a distance $av$ in a time interval of lenght $a$), then, since the length is bounded by $L_0$ and the curves have constant speed and are parametrized completely on $[0,1]$ this means the speed is bounded (from above) by $L_0$.