
Limit proof
Assume $\displaystyle f(x) >= g(x)$ for all $\displaystyle x$ in some set $\displaystyle A$ on which $\displaystyle f$ and $\displaystyle g$ are defined. Show that for any limit point $\displaystyle c$ of $\displaystyle A$ we must have $\displaystyle \displaystyle \lim_{x \to c} f(x) >= \lim_{x \to c} g(x).$
I think I start with the $\displaystyle \epsilon$ and $\displaystyle \delta$ definition of a limit. So let $\displaystyle \displaystyle \lim_{x \to c} f(x) = F.$ Let $\displaystyle \epsilon > 0.$ $\displaystyle 0 < x  c < \delta$ implies there exists $\displaystyle \delta > 0$ such that $\displaystyle f(x)  F < \epsilon.$
Is that the right place to start? Because I am not sure where to go from here.
Thanks in advance.

Let $\displaystyle \displaystyle \lim_{x \to c} f(x) =F~\&~ \lim_{x \to c} g(x)=G. $
Suppose that that $\displaystyle F<G$. You can get a contradiction.
Hint: consider $\displaystyle \varepsilon = \frac{{G  F}}{2}$.

Thank you so much. I believe I have it.