1. ## Diagonalization troubles.

Let T be the union of the Tks from k=1 to infinity. Use a diagonalization process like that one used to show the set of positive rational numbers is denumerable to show that T is denumerable.

I looked at the argument for the positive rationals, but can not decide what to make the entries that I enumerate here. I am a bit confused.

Thanks.

2. Originally Posted by zhupolongjoe
Let T be the union of the Tks from k=1 to infinity. Use a diagonalization process like that one used to show the set of positive rational numbers is denumerable to show that T is denumerable.
You will have to supply a great deal more information to make the question intelligible.
Like what are the $T_k$?

By “diagonalization process” do you mean the so-called zigzag argument?
I think of Cantor’s ‘diagonal argument’ as being used to show a set is non-denumerable.

3. My apologies, I forgot the first part of the problem...I got this part, but it is probably needed to answer the part I didn't get:

For all natural numbers n, let Tn be the set of all sequences of exactly n 1's. I showed this is denumerable for all n.

Now I need to use the diagonalization, which my book uses to show the positive rationals are denumerable.

4. I can show you a proof for that statement, but it is not a diagonalzation argument.
It involves the prime factorization theorem, mapping $\{T_n\}$ to the positive integers one-to-one.

5. Thanks, but I already did that part and now I just need a diagonalization argument.

6. From my understanding, the diagonal argument is used to prove a set is uncountable, not to prove it is countable.