From the book "Putnam and Beyond."
The problem:
Show that the interval [0, 1] cannot be partitioned into two disjoint sets A and B such that B = A + a for some real number a.
Proof:
Assume that A,B, and a satisfy A∪B =[0,1], A∩B =∅, B =A+a. We can assume that a is positive; otherwise, we can exchange A and B. Then (1 − a, 1] ⊂ B; hence (1 − 2a, 1 − a] ⊂ A. An inductive argument shows that for any positive integer n, the interval (1−(2n+1)a, 1−2na] is in B, while the interval (1−(2n+2)a, 1−(2n+1)a] is in A. However, at some point this sequence of intervals leaves [0, 1]. The interval of the form (1 − na, 1 − (n − 1)a] that contains 0 must be contained entirely in either A or B, which is impossible since this interval exits [0, 1]. The contradiction shows that the assumption is wrong, and hence the partition does not exist.
I don't really understand what's going on:
An inductive argument shows that for any positive integer n, the interval (1−(2n+1)a, 1−2na] is in B.
How did he come to this conclusion?
The interval of the form (1 − na, 1 − (n − 1)a] that contains 0 must be contained entirely in either A or B, which is impossible since this interval exits [0, 1].
The wording here is very confusing. Is he merely pointing to the fact that [0, 1] is a closed interval, while the other is open? — "This interval exists [0, 1]" is not a sentence that makes any syntactical sense to me.