Sum of Cauchy Sequences Cauchy?
A sufficient condition weaker than translation invariance is that translations are uniformly equicontinuous, i.e that for every $\epsilon > 0$ there is $\delta > 0$ such that for all $x, y, z$, $d(x,y) < \delta$ implies $d(x+z, y+z) < \epsilon$. For then if $\{a_n\}$ and $\{b_n\}$ are Cauchy sequences, take $N$ so for $n, m > N$, $d(a_n, a_m) < \delta$ and $d(b_n, b_m) < \delta$, and note that $d(a_n + b_n, a_m + b_m) \le d(a_n + b_n, a_m + b_n) + d(a_m + b_n, a_m + b_m) < 2 \epsilon$.