Hello,

You can use the fact that is countable. Which can be done by induction, and you'll have to use the fact that is countable.

1.

You know that there exists an injective mapping

Let

Basis : for n=2, it's verified (I assume it's a know fact for you)

Inductive hypothesis :assumethat there exists an injective mapping

Now, you have toprovethat there exists an injective mapping

For this, define this way :

It is easy to show that it's injective, knowing that is injective.

2.

Now prove that there exists an injection from to

Since are countable, there exist injections

Define

Once again, it's easy to show that it's injective.

3.

We know that are injective.

Now you just have to prove that the composite of two injective functions is injective, in particular .

And you're done.