for n<=6 $$ 1/2 - 3/n \le0 $$ so if you are taking integer value of n, put n = 7 it gives 1/14 this is how he arrived in the conclusion ie, for $$ n\ge7 $$ $$ 1/2 - 3/n \ge1/14 $$ because n is in denominator it will only decrease the value of -ve term so overall value will increase with n ,which is the constant c1.

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 $c_1$.