How can I show there's exist Bijection between Q x Q and natural numbers (Q × Q ~ N)
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Well, you could do this in two steps.
1) There is a bijection between $\Bbb Q$ and $\Bbb N$ by using the Cantor's first diagonal argument.
2) There is a bijection between $A$ and $A\times A$, where $A$ is a denumerable (i.e., countably infinite) set. Here you can use Cantor's first diagonal argument as well.
The composition of bijective mappings is a bijection. Done.
Wuestenfux
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