Hint 2.5.15.1.

For the inductive step, you can assume you have a strictly increasing sequence up to \(a_k\) where \(a_k \lt 100\text{.}\) Now you just need to find the next term \(a_{k+1}\) so that \(a_{k} \lt a_{k+1} \lt 100\text{.}\) What should \(a_{k+1}\) be?

in-context