Exercise 18.

Prove that there is a sequence of positive real numbers \(a_0, a_1, a_2, \ldots\) such that the partial sum \(a_0 + a_1 + a_2 + \cdots + a_n\) is strictly less than \(2\) for all \(n \in \N\text{.}\) Hint: think about how you could define what \(a_{k+1}\) is to make the induction argument work.