Exercise 15.

While the above proof does not work (it better not since the statement it is trying to prove is false!) we can prove something similar. Prove that there is a strictly increasing sequence \(a_1, a_2, a_3, \ldots\) of numbers (not necessarily integers) such that \(a_n \lt 100\) for all \(n \in \N\text{.}\) (By strictly increasing we mean \(a_n \lt a_{n+1}\) for all \(n\text{.}\) So each term must be larger than the last.)