MathDB

Let

a_1, a_2,\cdots , a_n

and

b_1, b_2,\cdots , b_n

be (not necessarily distinct) positive integers. We continue the sequences as follows: For every

i>n

a_i

is the smallest positive integer which is not among

b_1, b_2,\cdots , b_{i-1}

, and

b_i

is the smallest positive integer which is not among

a_1, a_2,\cdots , a_{i-1}

. Prove that there exists

N

such that for every

i>N

we have

a_i=b_i

or for every

i>N

we have

a_{i+1}=a_i

Problem Statement