Let
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
be a non-empty set of positive integers. Suppose that there are positive integers
![b_1,\ldots b_n](/media/m/e/4/0/e4094a0045af22d7f8a02e029bffcc0f.png)
and
![c_1,\ldots,c_n](/media/m/4/1/6/41667cc2b05a94ae9d7a12b1aa50c04d.png)
such that
- for each
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
the set
![b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\}](/media/m/3/c/d/3cdd229716ffa7ea836972fb90e90a5e.png)
is a subset of
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
, and
- the sets
![b_iA+c_i](/media/m/3/6/0/3609429e8bb4b50c53dc5acc8ecdd932.png)
and
![b_jA+c_j](/media/m/a/a/b/aabde3b0ccc7023016c7644445e66665.png)
are disjoint whenever
Prove that
%V0
Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_1,\ldots b_n$ and $c_1,\ldots,c_n$ such that
- for each $i$ the set $b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\}$ is a subset of $A$, and
- the sets $b_iA+c_i$ and $b_jA+c_j$ are disjoint whenever $i\ne j$
Prove that $${1\over b_1}+\,\ldots\,+{1\over b_n}\leq1.$$