Let $a_1,\dots ,a_r,b_1,\dots ,b_r\in \mathbb{N}$ such that ${\sum }_i\frac{1}{a_i}>1$. For any finite sequence of $n$ (not necessarily distinct) integers $A=(x_1,\dots ,x_n)$ let $T(A)$ denote the sequence of length $rn$ given by$$(a_i{x}_j+b_i{)}_{1\le j\le n,1\le i\le r}.$$Prove that, if $A_1=(1)$ and $A_{i+1}=T(A_i)$, then there must be some $A_k$ with repeated elements.