If you have a set where there's an idea of the distance between two points (a metric space), you can talk about sequences where the points get arbitrarily close together. We call these Cauchy sequences.
If a metric space has the property that every Cauchy sequence converges to a point in the space, we call it complete.
The original set of all 1/n isn't complete because the sequence 1/2, 1/3, 1/4,... could only converge to 0, which isn't in the set. If we add 0 in, the set becomes complete.
OP is saying their partner completes them :)
It's even more sweet because OP's set has the additional property that all possible Cauchy sequences have the same limit of 0, so there's only one possible limit - every end goal of every part of the set is the same, the 0. No other number will do.
I included this note on the flowers to be delivered to my SO:
Me: { 1/n : n ≥ 1 }
You: { 0 }
Us: Me ∪ You