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**ThePerfectHacker** What bothers me is that if $\displaystyle f\in S$ then $\displaystyle f = \{ (n,a_n) : n\geq 1,a_n\in \mathbb{R}, |a_1|<1,|a_3|<2,|a_4|<1\}$. Thus, $\displaystyle f$ is a set of ordered pairs. But if $\displaystyle S = ]-1,1[\times\mathbb{R}\times ]-2,2[\times ]-1,1[\times\mathbb{R}^{\mathbb{N}}$ then it means $\displaystyle f = (a_1,a_2,a_3,a_4,g)$ where $\displaystyle a_j \in \mathbb{R}, |a_1|<1,|a_3|<2,|a_4|<1$ and $\displaystyle g$ is a function $\displaystyle \mathbb{N}\to \mathbb{R}$. How can you have $\displaystyle f = (a_1,a_2,a_3,a_4,g)$?