My Question reads:
Use the Cantor–Schröder–Bernstein Theorem to prove the following.
The set of all integers whose digits are 6, 7, or 8 is denumerable
Now, my understanding of the definition of CSB Theorem is if cardinality of A less than or equal to cardinality B and cardinality B less than or equal to cardinality A, then cardinality A equals cardinality B.
I have started by setting S=the set of all integers whose digits are 6,7, or 8. From there I am not too sure what function to define. From this set S to what other set? And I know I would also have to show a function the other way around; in both cases one-to-one functions. I am more concerned as to how to go about setting this up. I am not too sure what is meant by integers whose digits are 6,7,or 8. Would this be a number like 67, or 86?
Updated Answer
I am working on defining f: Z->A by defining:
f(x)={6x, if x>0; 7x, if x=0; 8x, if x>0}
Can something like this be used to solve this direction?
Re your last question: yes.
If $A$ is your set, then we have the inclusion $A\to\Bbb N$, $x\mapsto x$ showing $|A|\le |\Bbb N|$ and we have $\Bbb N\to X$, $n\mapsto 6\cdot \frac{10^n-1}{9}$ showing $|\Bbb N|\le |A|$.