Is product of countably many copies of $\{0,1\}$ uncountable? Let $X$ denote the two point set $\{0,1\}$ and let $X_j=\{0,1\}\forall j=1,2\dots$ let $Y=\Pi_{j=1}^{\infty}X_j$, I need to determine whether each of the following are true or false:


*

*$Y$ is countable

*$|Y|$=|[0,1]|

*$\bigcup_{n=1}^{\infty}\Pi_{j=1}^{n}X_j$ is uncountable

*$Y$ is uncountable.
I guess $Y$ is uncountable (4), but I can not prove it.

HINTS


*

*On a meta-level, if the first is true then the other three are trivially false.

*Every real in $[0,1]$ has a binary decimal expansion $0.b_1b_2b_3\ldots$

