If for any sets $X$ and $Y$ one writes $Y^{X}$ to denote the set of all functions $f:X\to Y$ then shouldn't the set of all sequences with entries in a field $\mathbb{F}$ be given by the expression $S=\bigcup_{f\in \mathbb{F}^{\mathbb{N}}}\prod_{k=1}^{\infty}\{f(k)\}$ as expanding the Cartesian product of singletons in $S$ gives the following:
$$S=\bigcup_{f\in \mathbb{F}^{\mathbb{N}}}\prod_{k=1}^{\infty}\{f(k)\}=\bigcup_{f\in \mathbb{F}^{\mathbb{N}}}\{f(1)\}\times \{f(2)\}\times\{f(3)\}\times \cdots=\bigcup_{f\in \mathbb{F}^{\mathbb{N}}}\{(f(1),f(2),f(3),\cdots)\}\\=\{(f(1),f(2),f(3),\cdots):f\in \mathbb{F}^{\mathbb{N}}\}=\{(f(1),f(2),f(3),\cdots):\text{For all functions }f:\mathbb{N}\to \mathbb{F}\}$$
Yet according to various responses/answers on this website as well as Wikipedia people refer to the set of all sequences with entries in $\mathbb{F}$ as simply $\mathbb{F}^{\mathbb{N}}$: https://en.wikipedia.org/wiki/Sequence_space
How can this make sense? If every element of the set $f\in \mathbb{F}^{\mathbb{N}}$ is a function from $f:\mathbb{N}\to \mathbb{F}$ which in no way corresponds to a sequence, much less anything resembling say a tuple of elements in $\mathbb{F}$.
"A function... in no way corresponds to a sequence" - on the contrary, a sequence is nothing else but a function with domain $\Bbb N$. The notation $$a_0\,,\ a_1\,,\ a_2\,,\ldots$$ is merely another way of writing $$a(0)\,,\ a(1)\,,\ a(2)\,,\ldots\ .$$