Is there a counterexample for the claim in the question subject, that a sum of two closed sets in R is closed? If not, how can we prove it?

(By sum of sets X+Y I mean the set of all sums x+y where x is in X and y is in Y)

Thanks!

**Answer**

Consider the sets A={n∣n=1,2,…} and B={−n+1n∣n=2,3,…}. Note that 0 is not in the sum, but 1n is for each n≥2.

**Attribution***Source : Link , Question Author : ro44 , Answer Author : David Mitra*