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Proposition: Preservation of Inequalities for Limits of Functions

Let $D\subseteq\mathbb R$ be a subset of real numbers, $a\in\mathbb R$ be a real number, and let $f:D\to\mathbb R$ be a function with the limit $\lim_{x\to a}f=L.$ Then,

• If $f(x) > 0$ for all $x\in D,$ then $L \ge 0.$
• If $f(x) < 0$ for all $x\in D,$ then $L \le 0.$

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References

Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016