Why is there a change in inequality from ">" to "$\geq$" in this example? I copied the example out but I am not interested in the example per say, except for the change in inequality that takes place. I underlined it in red.
The Example

My Question
What I understand: (for RHS)
$f(x) - g(x) < \frac{f(p)}{2}$
$\implies g(x) >f(x) - \frac{f(p)}{2}$
so how did get "$\geq$"?
 A: Consider two numbers $x$ and $y$. Then $x\geq y$ implies either $x=y$ OR $x>y$ holds.
So, if $x>y$ is true then so is $x\geq y$ since $x\geq y$ requires at least one of $x=y$ OR $x>y$ to be true. Think of it in terms of (mathematical) logic.
Thus, if the statement $g(x)>f(x)-\dfrac{f(p)}{2}$ is true, then this implies that the statement $g(x)\geq f(x)-\dfrac{f(p)}{2}$ is also automatically true.
A: As already mentioned in previously submitted answer, logically you can say $x > y$ can also be written as $x ≥ y$ because at least one of the cases is valid. According to this $5 ≥ 2$ also taken as correct. But I think it's better to include equal sign if there is a possibility for it to happen because it is a special case. 
Let's consider an integer $n$, can you say $n > 5$ is same as $n ≥ 5$? I suggest you can't because in the first case minimum of $n$ is $6$, but in the second case minimum of $n$ is $5$, therefore the implications are totally different. 
Let's consider another situation,  
Does it imply $a ≥ c$  given that $a ≥ b > c$ or $a ≥ c$ given that $a ≥ b ≥ c$?
