(related to Proposition: Preservation of Inequalities for Limits of Functions)

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

```
* Hypothesis:
* Assume `$L < 0.$`
* Implications
* By the [definition of limit][bookofproofs$6683], for `$\epsilon=-L$` there is a `$\delta > 0$` such that for all `$x\in D$` satisfying `$0 < |x-a| < \delta,$` it follows that `$|f(x)-L| < - L.$`
* For these values of `$x$` we have `$-L > |f(x)-L| = f(x) - L,$` implifying that `$f(x) > 0$` and `$L < 0.$`
* This creates the [contradiction][bookofproofs$744] to `$f(x) < 0.$`
* Conclusion
* It must be that `$L\ge 0.$`
```

```
* Hypothesis:
* Assume `$L > 0.$`
* Implications
* By the [definition of limit][bookofproofs$6683], for `$\epsilon=L$` there is a `$\delta > 0$` such that for all `$x\in D$` satisfying `$0 < |x-a| < \delta,$` it follows that `$|f(x)-L| < L.$`
* For these values of `$x$` we have `$L > |f(x)-L| = -f(x) + L,$` implifying that `$f(x) < 0$` and `$L > 0.$`
* This creates the [contradiction][bookofproofs$744] to `$f(x) > 0.$`
* Conclusion
* It must be that `$L\le 0.$`<div class='qed'>∎</div>
```

**Kane, Jonathan**: "Writing Proofs in Analysis", Springer, 2016