Cardinality of a set of functions from $\mathbb N$ to a set of real numbers

Question

Let $S$ be the set of functions from $\mathbb N$ to the set $\{\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}\}$. Determine $|S|$.

I understand how to prove the converse case where $T$ is the set of functions from $\{\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}\}$ to $\mathbb N$ by defining a bijective function $f:T \to \mathbb N \times \mathbb N \times \mathbb N \times \mathbb N$, where $|\mathbb N \times \mathbb N \times \mathbb N \times \mathbb N| = |\mathbb N|$. How would I approach this case though?

Answer 1

Do you know how to calculate the cardinality of all $0$-$1$ sequences?