About the properties of the coresets in k-median clustering
The k-medians clustering objective is defined here as
sum_{p in points} weight(p) distance(p, centers)

where centers is the set of k centers and distance(p, centers) is the distance to the closest center.
The first observation amounts to observing that the objective function is additive. Letting (C1, w1) and (C2, w2) be the core-sets, we consider the core-set (C1 union C2, w1 union w2), which satisfies
sum_{p in C1 union C2} (w1 union w2)(p) distance(p, centers)
= sum_{p in C1} w1(p) distance(p, centers) + sum_{p in C2} w2(p) distance(p, centers)
&lt;= exp(eps) sum_{p in P1} weight(p) distance(p, centers) + exp(eps) sum_{p in P2} weight(p) distance(p, centers)
= exp(eps) (sum_{p in P1 union P2} weight(p) distance(p, centers))

and similarly for the lower bound.
As for the second observation,
sum_{p in C1} w1(p) distance(p, centers)
&lt;= exp(eps) sum_{p in C2} w2(p) distance(p, centers)
&lt;= exp(eps) exp(delta) sum_{p in C3} w3(p) distance(p, centers)
= exp(eps + delta) sum_{p in C3} w3(p) distance(p, centers),

and similarly for the lower bound.