Is product of countably many copies of $\{0,1\}$ uncountable?

Question

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:

1. $Y$ is countable

2. $|Y|$=|[0,1]|

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

4. $Y$ is uncountable.

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

Answer 1

HINTS

1. On a meta-level, if the first is true then the other three are trivially false.
2. Every real in $[0,1]$ has a binary decimal expansion $0.b_1b_2b_3\ldots$

Answer 2

Every $a \in Y$ is an infinite series of zeros and ones. Think of any $b \in P(\mathbb N)$ and try to find the connection between this two.