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.)
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