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Thread: Limit proof

  1. #1
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    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.
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  2. #2
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    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 $\displaystyle \varepsilon = \frac{{G - F}}{2}$.
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  3. #3
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    Thank you so much. I believe I have it.
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