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partition of set of all finite ordered sets of 0 and 1

Source: I Soros Olympiad 1994-95 Ukraine R2 11.2 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

June 6, 2024

combinatorics

Problem Statement

The set of all finite ordered sets of

0

and

1

is somehow partitioned into two disjoint classes. Prove that any infinite sequence of

0

and

1

can be cut into non-intersecting finite parts such that all of these parts (except perhaps the first) belong to the same class.