Group covered by finitely many cosets This question appears in my textbook's exercises, who can help me prove it?

If a group $G$ is the set-theoretic union of finitely-many cosets, $$G=x_1S_1\cup\cdots\cup x_nS_n$$ prove that at least one of the subgroups $S_i$ has finite index in $G$.

I think that the intersection of these cosets is either empty or a coset of the intersection of all the $S_i$. I want to start from this point to prove it. So I suppose none of these $S_i$ has finite index, but I don't know how to continue?

I believe that this is a nontrivial result. It was proven in 

B. H. Neumann, Groups covered by finitely many cosets, Publ. Math.
  Debrecen 3 (1954), 227–242. MR 17, 234.

Unfortunately, I could not find the original argument. Perhaps someone with a sharper Google-fu can retrieve it.
(I am curious to know what book you found this in. It seems like a cruel exercise, at least assuming that your book is an introductory book on group theory.)
