1. ## Proving countable set

I need help proving that the set: (1/2, 2/3, 3/4,...,n/n+1,...) is countable.

2. Originally Posted by redwings6
I need help proving that the set: (1/2, 2/3, 3/4,...,n/n+1,...) is countable.
Let $\displaystyle G = \left\{ {\frac{n}{{n + 1}}:n \in \mathbb{Z}^ + } \right\}$.
Now define $\displaystyle \Phi:G\mapsto \mathbb{N}$ by $\displaystyle \Phi \left( {\frac{n}{{n + 1}}} \right) = n$.

Prove that $\displaystyle \Phi$ is injective.

3. Originally Posted by redwings6
I need help proving that the set: (1/2, 2/3, 3/4,...,n/n+1,...) is countable.
Alternatively, clearly $\displaystyle \eta:\mathbb{N}\to K:n\mapsto\frac{n}{n+1}$ is an overkill surjection.

Clearly Plato's answer along with "every infinite subset of a countable set is countable" is better though.