Proof that the set of  odd positive integers  greater then 3 is countable I found  one  problem which  asks  following:

Show that the set of  odd positive integers  greater then 3 is countable.

At the begining I was thinking that  such numbers  could be represented   by $2k+1$,where $k>1$; but  in the answers paper  there was  written  as $2n+3$  or  general function   is
$$f(n)=2n+3$$
and  when I was thinking how to prove that  such answer  is countable, the answer  paper said  this function is a one-to-one correspondence  from the set of positive  numbers set to the set of positive odd  integers  greater to 3. 
My  question is:  is it  enough  to prove a one-to-one  correspondence  between  two sets, that one of them  is  countable. If yes, then once my lecturer ask me to proof that rational numbers are countable, so in this case if I represent  rational numbers  by following  function from set of positive numbers:
$$f(n)=\frac{n+1}{n}$$
 or  maybe   $f(n)=\frac{n}{n+1}$. They both are one-to-one correspondences from the set of positive numbers to the set of rational numbers (positives sure). Please help me, is my logic correct or not?

The sufficient condition for proving countability is establishing the 1-1 correspondence of a given set with $\mathbb{N}$. For proving the countability of rationals, a method called as diagonalization is used. Take a look at the following link. Your logic is correct, only that in place of $f(x)=\frac{n}{n+1}$, we use $f(x)=\frac{m}{n}$ and express $\frac{m}{n}$ in mth row and nth column. The representation covers all possible rational numbers and their ordering makes it a set corresponding 1-1 to $\mathbb{N}$.
http://www.homeschoolmath.net/teaching/rational-numbers-countable.php
