Are the $U_n$ supposed to have a physical meaning or is this a purely mathematical exercises? if we directly interpret:

$$ U_n = M\left(c-\frac{1}{n+2}\right)-M\left(c-\frac{1}{n+1}\right)\label{2}\tag{$U$} = \frac{m_0}{\sqrt{1 - 1 + \frac{2}{c(n+2)} - \frac{1}{c^2(n+2)^2}}} - \frac{m_0}{\sqrt{1 - 1 + \frac{2}{c(n+1)} - \frac{1}{c^2(n+1)^2}}}
= m_0 \left( \frac{1}{\sqrt{\frac{2}{c(n+2)} - \frac{1}{c^2(n+2)^2}}} - \frac{1}{\sqrt{\frac{2}{c(n+1)} - \frac{1}{c^2(n+1)^2}}} \right) = m_0 c \left( \frac{n+2}{\sqrt{2c(n+2) -1 }} - \frac{n+1}{\sqrt{2c(n+1)-1}} \right) = $$

It's not clear to me how to assign units to the last object. It seems like the $n$ and $1$ and $2$ need to be given units for the whole thing to remain physical with dimensional analysis.

Anyways... I believe that $U_n = O(n^{-\frac{1}{2}}) $ so one could use the Euler Maclaurin formula on this and see if that suggests a constant term in its asymptotic expansion. This will be very tedious and I'll get around to it later (unless someone beats me to it :) )