Trouble understanding how to pick constants to prove big theta
For $n \leq 6$, 
$$
1/2 - 3/n \le0,
$$
so if you are taking integer value of $n$, put $n = 7$ as it gives $1/14$. This is how he arrived in the conclusion i.e., for $n \geq 7$, 
$$ 
1/2 - 3/n \ge1/14,
$$
because $n$ is in the denominator it will only decrease the value of -ve term, so overall value will increase with $n$, which is the constant $c_1$.