Set of finite and infinite $0$-$1$ sequences countability

Question

Possible Duplicate:
Is the set of all finite sequences of letters of Latin alphabet countable/uncountable? How to prove either?
Is the set of all strings with countably infinite length bijective to $\[0,1\]$?

I'm trying to prove that a set of a finite sequences of $0,1$ (let's call it $A$) is countable infinite, whereas a set of infinite sequences of $0,1$ (call it $B$) equipotent to $P(\mathbb N)$ is uncountable infinite.

So far I tried showing that the number of sequences possible in A is $\sum \limits_{i=1}^n \ 2^i$. Not sure how to continue from here, if this is even the right direction.

Answer 1

I'll do the first part:

For each positive integer $n$, let $A_n$ be the set of sequences of 0's and 1's of length $n$. Then $A=\bigcup_{n=1}^\infty A_n$.

Moreover, each $A_n$ is finite. In fact $|A_n|=2^n$. Since a countable union of countable sets is countable, it follows that $A$ is countable.

Now the sequence $\underbrace{0\ 0\ 0\ \cdots\ 0 }_{n-\rm terms}\ 1$ is in $A$ for each positive integer $n$. Thus, $A$ is infinite (and so, countably infinite).

Answer 2

B is a standard application of Cantor's diagonal argument, though the obvious bijection is far more enlightening.