Cheers everyone. I'm going through a paper by Paul Erdos and am slightly flummoxed by one of the steps in his proof. Here goes:

Let and be residual sets in (in the sense of Baire category). For a fixed real number , the sets and are also residual.

I can see why this is true intuitively, particularly for , since a simple linear transformation of the elements of isn't going to effect the category of the complementary set, but I can't seem to drum up a rigorous proof. No doubt it is going to involve the nowhere density of related sets, but I can't seem to connect the dots. Any help would be greatly appreciated. Many thanks.