Question 21
There is an infinite sequence that uses the digits $1$, $2$, $\dots$, $9$. We consider each sequence of consecutive digits as a positive number written in base $10$.
Show that for anyone including $n \ge 2$ at least one of the following statements is true:
(i) If you can find $n$ numbers $A_1, A_2, \dots, A_n$ formed by consecutive digits in succession, each one strictly according to the previous one, each one with $n$ digits and as many as $A_1 \lt A_2 \lt \dots \lt A_n$.
(ii) The sequence contains a number of $n$ to a maximum of $n-1$ digits that are repeated consecutively for at least $n+2$ times.
Show that for anyone including $n \ge 2$ at least one of the following statements is true:
(i) If you can find $n$ numbers $A_1, A_2, \dots, A_n$ formed by consecutive digits in succession, each one strictly according to the previous one, each one with $n$ digits and as many as $A_1 \lt A_2 \lt \dots \lt A_n$.
(ii) The sequence contains a number of $n$ to a maximum of $n-1$ digits that are repeated consecutively for at least $n+2$ times.