Come on, do not take a sledgehammer to crack a nut. There is a simple example:
Let the be such infinite set that , where is the i-th prime number and .
Then infinite family is an example satisfying the requirements. Because for every distinct and and there are infinitely many primes.
I read a problem like this somewhere.
since there is a bijective mapping between N and Q, let A be the collection of all rational number cauchy sequence converging to 0, B be the collection of all rational number cauchy sequence converging to 1.
A union B is a uncountable set. for each a in A, and b in B, there are only finite common elements.