You're going to be told to make another thread.

Also, why did you make me go get my book? haha

Question:

The following reflects my own personal taste and might be unappealing to you.

So, if

this is trivial. So, assume not. Then, as you've shown

where each

. So, notice that

but each

is the countable union of countable sets

and thus countable. So, for each

there exists some surjection

and so define

where

is the unique element of

such that

.

To see this is a surjection we merely note that if

then

for some

and so there exists some

such that

. The conclusion follows.

Note the absolute necessity in the above that

and not