Do we need Axiom of Choice to make infinite choices from a set? According to the answers to this question, we do not need choice to pick from a finite product of nonempty sets, even if each of the sets is infinite. The axiom of choice is required to ensure that a infinite product of nonempty sets is non-empty. i.e. $\prod_{i \in I} A_i \neq 0$.
Now, let $A_i = \mathbb{R}$. The answers to this question (and the one linked above) says we do not need choice to pick an element $x_0 \in \mathbb{R}$. Suppose, I want an arbitrary sequence of real numbers $X = (x_n)_{n =1}^{\infty}$. Then, I will have to make an infinite number of "picks" from $\mathbb{R}$.
Is it right to say that the resulting sequence $X \in \prod_{i \in I} \mathbb{R}$ and that we need choice to ensure that it exists? Why or why not?

No. You may simply pick $x_i=42$ for all $i\in I$.
