Sum of two closed sets in R\mathbb R is closed?

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={nn=1,2,} and B={n+1nn=2,3,}. Note that 0 is not in the sum, but 1n is for each n2.

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

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