# 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 "$$\geq$$"?

• Surely $A > B$ implies $A \geq B.$ I don't see any particular reason why it's not "$>$" in the second underlined inequality, but writing $\geq$ doesn't spoil the proof. Jun 27, 2022 at 2:02
• @DavidK, thank you for your reply. But shouldn't $A > B$ mean that $A$ never actually $= B$, but $A \geq B$ implies that $A$ can $= B$. Allowing this feels like room for some misplay, or am I not fully understanding what "$A > B$" really means? Jun 27, 2022 at 2:08
• I see there is a full answer now, so I don't need to explain. But for what it's worth, I think $>$ would have been preferable to $\geq$. It just happens that the proof still holds up, but it introduces an extraneous "or $g(x)=f(x)-\frac{f(p)}{2}$" which we know has no effect since $g(x)\neq f(x)-\frac{f(p)}{2}$, and the proof would be cleaner if this extraneous "or" weren't there. Jun 27, 2022 at 2:31
• If "Adam is an anteater" is true, then "Adam is an anteater or a zebra" is also true. "Adam is an anteater" is a stronger statement because it is telling you that Adam is most assuredly and anteater and the second is only telling you that "Adam is either an anteater or a zebra". But a weaker statement is still a true statement... After all if it were not true that $A \ge B$ it what have to be true that $A < B$ and that is WRONG. So $A \ge B$ is true. Jun 27, 2022 at 2:42
• Actually, I bet that was just a type-setting error that never was caught. I bet the manuscript did say $g(x) > f(x) - \frac {f(p)}2$. (But what is written is not technically wrong). Jun 27, 2022 at 2:44

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.
• For example, it is true to say that $1 \geq 2$, even though it would be a strange thing to say Jun 27, 2022 at 6:27
• @PrinceM Do you mean $1\leq 2$? Jun 27, 2022 at 6:42
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.
Does it imply $$a ≥ c$$ given that $$a ≥ b > c$$ or $$a ≥ c$$ given that $$a ≥ b ≥ c$$?