$X=\mathbb{N}^\mathbb{N}$ itself has cardinality $\mathfrak c$ (the continuum), by the same digit-sequence argument used for finite-alphabet sequences. Any subfamily that still leaves infinitely many coordinates free to vary independently over $\mathbb{N}$ — such as " $x_n=0$ infinitely often" (only forcing zeros along some subsequence, with the rest unconstrained) — remains uncountable, since one can already build a continuum-sized family by fixing the zero pattern and varying the remaining coordinates freely.

$W$ , by contrast, consists of sequences that are eventually identically zero: $x_n\neq0$ for only finitely many $n$ . Such a sequence is completely determined by a finite index $N$ (beyond which everything is $0$ ) together with finitely many values before it, each drawn from $\mathbb{N}$ — a countable union (over $N$ ) of countable sets (each a finite Cartesian power of $\mathbb{N}$ ), hence countable. This is the classical fact that "eventually zero" sequences form a countable set inside the uncountable $\mathbb{N}^\mathbb{N}$ .