A k-term combinatorial progression of order d (abbreviated as k-CP(d)) is defined as an integer sequence $x_1
A m-cube is defined as a set of the form $=\{a+\sum_{i=1}^m e_iy_i : e_i \in \{0,1\}\}$. We say that a set of positive integers A has the property C if A contains an m-cube $\forall m\ge 1$.
I am trying to establish that $CP\implies C$. The author of the paper I am reading from says it is sufficient to prove the statement: For all $m,d\ge 1$ there exists $r=r(d,m)$ such that if $x_1,x_2\cdots x_r$ is an r-CP(d) then $\{x_1,x_2\cdots x_r\}$ contains an m-cube.
My problem is why is this sufficient to establish this statement? The point I am stuck on is that we are guaranteed a combinatorial progression of a desired length but not guaranteed a combinatorial progression of the desired order. Hence in searching for an m-cube we may only use CP to get r-CP(d') where d' is bigger then d and hence the hypothesis (requisite r-CP(d) forces an m-cube) may not be invoked.
In case anyone is interested this is from an Erdos paper where the author remarks that it is easy to see this. I can't find it easy at all and will be obliged if someone guides me.
Thanks a lot.