# Limit proof

• Nov 6th 2010, 03:19 PM
Zalren
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.

• Nov 6th 2010, 03:36 PM
Plato
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 $\displaystyle \varepsilon = \frac{{G - F}}{2}$.
• Nov 6th 2010, 04:10 PM
Zalren
Thank you so much. I believe I have it.