Comparing cardinalities Why these two sets are equinumerous?
$$[0,1]^\Bbb N\text{ and }\Bbb Q^\Bbb N$$
Here is my reason:
The set of rational numbers $\Bbb Q$ is countably infinite. However, $[0, 1]$ is not countable and is infinite.
So, they shouldn't be equinumerous. 
Even, there is the power of $\Bbb N$, it shouldn't change anything.
But, I am wrong.
Can anybody tell me what is wrong please?
Thank you in advance!

First of all, note that $\Bbb{Q^N}$ includes $\{0,1\}^\Bbb N$, so it too is uncountable. But just being uncountable doesn't mean much because there are uncountable sets of different cardinalities.
But note that $|[0,1]|=2^{\aleph_0}$ and $|\Bbb Q|=\aleph_0$. Therefore $[0,1]^\Bbb N$ has cardinality $(2^{\aleph_0})^{\aleph_0}$, and $\Bbb{Q^N}$ has cardinality $\aleph_0^{\aleph_0}$.
What do you know about these two cardinalities?
