ok, here it goes... Critique me, please. and keep in mind that I'm new at this.

"Prove that the set A of all sequences consisting only of zeros and ones has the power of the continuum."

To show that

has the power of the continuum, we will establish a one-to-one correspondence between

and

. First consider two subsets of

, which we will call

and

.

consists of all sequences having a finite number of zeros.

consists of all sequences having an infinite number of zeros.

The subset

is countable. Take a countable subset of

, say

, which we will call

. We can form a bijection between this set and

, call it

.

Now rewrite every sequence in

as a number written in base 2, keeping the number and order of the ones and zeros the same. Clearly there is a one-to-one correspondence between these numbers and

.

Rewrite

as

and

as

. Define our bijection as

whenever a sequence

and as rewriting the terms of

as a number in base 2 whenever

.

Since any subset of

is equivalent (denoted

) to the interval

(that is, it has the power of the continuum), we have that

. So, we have

and

, so

by transitivity and has the power of the continuum.