Bijection between products of countable sets

[The1TL]

Homework Statement

Let S1 = {a} be a set consisting of just one element and let
S2 = {b, c} be a set consisting of two elements.

Show that S1 × Z is bijective to S2 × Z.

Homework Equations

The Attempt at a Solution

So I usually prove bijectivity by showing that two sets are equinumerous, But in this case S1 and S2 are not so that makes it more difficult.

 

[spamiam]

You're right, this problem demonstrates how things can become unintuitive when dealing with infinite sets. If A were any nonempty finite set, this claim would be false, since in that case
$$|S_1 \times A| = |S_1||A| = |A|$$
$$|S_2 \times A| = |S_2||A| = 2|A| \; .$$

However, since $\mathbb{Z}$ is infinite, we have more leeway. Can you think of a proper subset of $\mathbb{Z}$ that is equinumerous to $\mathbb{Z}$?

 

[Deveno]

here is one idea:

...
...
(a,-3) <--> (c,-2)
(a,-2) <--> (b,-1)
(a,-1) <--> (c,-1)
(a,0) <--> (b,0)
(a,1) <--> (c,0)
(a,2) <--> (b,1)
(a,3) <--> (c,1)
(a,4) <--> (b,2)
...
...

can you prove this is, in fact, a bijection?