Question

When will sum of infimum be equal to infimum of sum?

Answer 1

First, let a=infA, b=infB. Pick x+y∈A+B. Then

a≤x,b≤y⟹a+b≤x+y

Second, let ϵ>0 be given. By definition of the infimum, there exists x in A and y in B such that

x<infA+ϵ2

y<infB+ϵ2

By summing

x+y<infA+infB+ϵ

The first step shows

∀z∈A+B we have infA+infB≤z

The second step shows

∀ϵ>0∃z∈A+B such that z<infA+infB+ϵ

This is precisely the (equivalent) definition of inf(A+B).